Gibbs measures with multilinear forms

Sohom Bhattacharya S. Bhattacharya Department of Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08540, USA. [email protected] , Nabarun Deb N. Deb Econometrics and Statistics
University of Chicago Booth School of Business
Chicago, IL 60637, USA. [email protected] and Sumit Mukherjee S. Mukherjee Department of Statistics
Columbia University
New York, NY 10027, USA. [email protected]

Abstract.

In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$ -statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain necessary and sufficient conditions for replica-symmetry. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the “local fields/magnetization”, the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^{n}c_{i}X_{i}\to 0$ weakly, if $\sum_{i=1}^{n}c_{i}=o(n)$ . Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest.

Key words and phrases:

Graph limits, magnetization, phase transition, replica-symmetry, tensor Ising model

1991 Mathematics Subject Classification:

82B20, 05C80

The third author’s research is partially supported by NSF grant DMS-2113414.

1. Introduction

Suppose $\mu$ is a (non-degenerate) probability measure on $\mathcal{\mathbb{R}}$ . Let $H=(V(H),E(H))$ be a finite graph with $v:=|V(H)|\geq 2$ vertices labeled $[v]=\{1,2,\ldots,v\}$ , and maximum degree $\Delta$ . Fixing $\theta\in\mathbb{R}$ , define a function

(1.1)

\displaystyle Z_{n}(\theta):=\frac{1}{n}\log\mathbb{E}_{\mu^{\otimes n}}e^{n\theta\mathbb{U}_{n}({\bf X})}\in(-\infty,\infty],

where $\mathbb{U}_{n}({\bf X})$ be a multilinear form, defined by

(1.2)

\displaystyle\mathbb{U}_{n}({\bf X}):=\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in\mathcal{S}(n,v)}\Big{(}\prod_{a=1}^{v}X_{i_{a}}\Big{)}\prod_{(a,b)\in E(H)}Q_{n}(i_{a},i_{b}).

Here $\mathcal{S}(n,v)$ is the set of all distinct tuples from $[n]^{v}$ (so that $|\mathcal{S}(n,v)|=v!\binom{n}{v}$ ), and $Q_{n}$ is a symmetric $n\times n$ matrix with $0$ on the diagonal. If $\theta$ is such that $Z_{n}(\theta)$ is finite, we can define a Gibbs probability measure $\mathbb{R}_{n,\theta}$ on $\mathbb{R}^{n}$ by setting

(1.3)

\displaystyle\frac{d\mathbb{R}_{n,\theta}}{d\mu^{\otimes n}}({\bf x})=\exp\Big{(}n\theta\mathbb{U}_{n}({\bf x})-nZ_{n}(\theta)\Big{)}.

Several Gibbs measures of interest can be expressed in the form (1.3) with various choices of $(Q_{n},H,\mu)$ . Below we give two examples of such Gibbs measures which have been well studied in Probability and Statistics.

•

If $H=K_{2}$ is an edge, then

\frac{d\mathbb{R}_{n,\theta}}{d\mu^{\otimes n}}({\bf x})=\exp\left(\frac{1}{n}\sum_{i\neq j}Q_{n}(i,j)x_{i}x_{j}-nZ_{n}(\theta)\right)

is a Gibbs measure with a quadratic Hamiltonian. In particular if $\mu$ is supported on $\{-1,1\}$ , then $\mathbb{R}_{n,\theta}$ is the celebrated Ising model on $\{-1,1\}^{n}$ with coupling matrix $Q_{n}$ (see [24, 3, 12, 1] for various examples). Popular examples of $Q_{n}$ include the adjacency matrix of the complete graph, line graph, random graphs such as Erdős-Rényi or random $d$ -regular, and so on.

•

If $H=K_{v}$ is the complete graph on $v$ vertices, and $Q_{n}$ is the adjacency matrix of a complete graph, then model (1.3) reduces to

\frac{d\mathbb{R}_{n,\theta}}{d\mu^{\otimes n}}({\bf x})=\exp\left(\frac{1}{n^{v-1}}\sum_{(i_{1},\ldots,i_{v})\in\mathcal{S}_{n,v}}\prod_{a=1}^{v}x_{i_{a}}-nZ_{n}(\theta)\right).

For the special case where $\mu$ is supported on $\{-1,1\}$ , $\mathbb{R}_{n,\theta}$ is just the $v$ -spin version of the Curie-Weiss model, which has attracted attention in recent years (see [29, 28, 33, 15]).

In this paper, we study the generalized model (1.3), when the sequence of matrices $\{Q_{n}\}_{n\geq 1}$ converge in weak cut metric (defined by (1.4)). Our main contributions are:

(a) We give an exact characterization for replica-symmetry for the asymptotic free energy/log partition function (see 1.2).

(b) We obtain weak limits for a large family of statistics which include the Hamiltonian, “local magnetizations”, global magnetization, and contrasts (see Theorems 1.3 and 1.6).

(d) We show the existence of a “phase transition” for multilinear Gibbs measures of the form (1.3) with compactly supported $\mu$ (see 1.9).

1.1. Main results

To establish our main results, we will assume throughout that the sequence of matrices $\{Q_{n}\}_{n\geq 1}$ converges in the weak cut distance (defined below). Cut distance/cut metric has been introduced in the combinatorics literature to study limits of graphs and matrices (see [21]), and have received significant attention in the recent literature ([8, 9, 10, 11]). For more details on cut metric and its manifold applications, we refer the interested reader to [27]. Below we formally introduce the notion of strong and weak cut distances used in this paper.

Definition 1.1.

Suppose $\mathcal{W}$ is the space of all symmetric real-valued functions in $L^{1}([0,1]^{2})$ . Given two functions $W_{1},W_{2}\in\mathcal{W}$ , define the strong cut distance between $W_{1},W_{2}$ by setting

d_{\square}(W_{1},W_{2}):=\sup_{S,T}\Big{|}\int_{S\times T}\Big{[}W_{1}(x,y)-W_{2}(x,y)\Big{]}dxdy\Big{|}.

In the above display, the supremum is taken over all measurable subsets $S,T$ of $[0,1]$ . Define the weak cut distance by

\delta_{\square}(W_{1},W_{2}):=\inf_{\sigma}(W^{\sigma}_{1},W_{2})=\inf_{\sigma}(W_{1},W^{\sigma}_{2})

where $\sigma$ ranges from all measure preserving bijections $[0,1]\rightarrow[0,1]$ and $W^{\sigma}(x,y)=W(\sigma(x),\sigma(y))$ .

Given a symmetric matrix $Q_{n}$ , define a function $W_{Q_{n}}\in\mathcal{W}$ by setting

\displaystyle W_{Q_{n}}(x,y)=

\displaystyle Q_{n}(i,j)\text{ if }\lceil nx\rceil=i,\lceil ny\rceil=y.

We will assume throughout the paper that the sequence of matrices $\{Q_{n}\}_{n\geq 1}$ introduced in (1.2) converge in weak cut distance, i.e. for some $W\in\mathcal{W}$ ,

(1.4)

\displaystyle\delta_{\square}(W_{Q_{n}},W)\rightarrow 0.

We now introduce some notation that will be used throughout the rest of the paper.

Definition 1.2.

Let $\mathcal{M}$ denote the set of probability measures on $[0,1]\times\mathbb{R}$ , equipped with weak topology. Given a probability measure $\nu\in\mathcal{M}$ , let $\nu_{(1)}$ and $\nu_{(2)}$ denote its first and second marginals respectively. Also define $\mathfrak{m}_{p}(\nu):=\int|x|^{p}\,d\nu_{(2)}(x)$ for $p\geq 0$ . Define $\widetilde{\mathcal{M}}\subseteq\mathcal{M}$ as follows:

\widetilde{\mathcal{M}}:=\{\nu\in\mathcal{M}:\ \nu_{(1)}=\mathrm{Unif}[0,1]\}.

Also define $\widetilde{\mathcal{M}}_{p}\subseteq\widetilde{\mathcal{M}}$ as follows:

(1.5)

\displaystyle\widetilde{\mathcal{M}}_{p}:=\{\nu\in\widetilde{\mathcal{M}}:\ \mathfrak{m}_{p}(\nu)<\infty\}.

Note that $\widetilde{\mathcal{M}}_{p}$ is a closed subset of $\widetilde{\mathcal{M}}$ (by Fatou’s Lemma), and $\widetilde{\mathcal{M}}$ is a closed subset of $\mathcal{M}$ , in the weak topology. For two measures $\nu_{1},\nu_{2}$ on $[0,1]\times\mathbb{R}$ , define

d_{\ell}(\nu_{1},\nu_{2}):=\sup_{f\in\text{Lip}(1)}\Bigg{|}\int fd\nu_{1}-\int fd\nu_{2}\Bigg{|},

where the supremum is over the set of functions $f:[0,1]\times\mathbb{R}\mapsto[-1,1]$ which are $1$ -Lipschitz.

We now introduce the exponential tilt of the base measure $\mu$ , and some related notations. This requires the following assumption, which we make throughout the paper: For all $\lambda>0$ and some $p\in[1,\infty]$ , we have

(1.6)

\displaystyle\mathbb{E}_{\mu}e^{\lambda|X_{1}|^{p}}<\infty,

where the case $p=\infty$ corresponds to assuming $\mu$ is compactly supported.

Definition 1.3.

Given (1.6), the function

\alpha(\theta):=\log\int_{\mathbb{R}}e^{\theta x}d\mu(x)

is finite for all $\theta\in\mathbb{R}$ . Define the $\theta$ -exponential tilt of $\mu$ by setting

\frac{d\mu_{\theta}}{d\mu}(x):=\exp(\theta x-\alpha(\theta)).

Then the function $\alpha(.)$ is infinitely differentiable, with

\alpha^{\prime}(\theta)=\mathbb{E}_{\mu_{\theta}}(X),\quad\alpha^{\prime\prime}(\theta)=\mathrm{Var}_{\mu_{\theta}}(X)>0.

Consequently the function $\alpha^{\prime}(.)$ is strictly increasing on $\mathbb{R}$ , and has an inverse $\beta(.):\mathcal{N}\mapsto\mathbb{R}$ , where $\mathcal{N}:=\alpha^{\prime}(\mathbb{R})$ is an open interval. Let $\mathrm{cl}$ denote the closure of a set in $\mathbb{R}$ , and extend $\beta(.)$ to a (possibly infinite valued) function on $\mathrm{cl}(\mathcal{N})$ by setting

	$\displaystyle\beta(\sup\{\mathcal{N}\})=$	$\displaystyle+\infty\text{ if }\sup\{\mathcal{N}\}<\infty,$
	$\displaystyle\beta(\inf\{\mathcal{N}\})=$	$\displaystyle-\infty\text{ if }\inf\{\mathcal{N}\}>-\infty.$

We write $D(\cdot|\cdot)$ to denote the standard Kullback-Leibler divergence. Define a function $\gamma:\beta(\mathrm{cl}(\mathcal{N}))\mapsto[0,\infty]$ by setting

$\displaystyle\gamma(\theta):=$	$\displaystyle D(\mu_{\theta}\\|\mu)=\theta\alpha^{\prime}(\theta)-\alpha(\theta)$	$\displaystyle\text{ if }\theta\in\mathbb{R}=\beta(\mathcal{N}),$
$\displaystyle\gamma(\infty):=$	$\displaystyle D(\delta_{\sup\{\mathcal{N}\}}\|\mu)$	$\displaystyle\text{ if }\sup\{\mathcal{N}\}<\infty,$
$\displaystyle\gamma(-\infty):=$	$\displaystyle D(\delta_{\inf\{\mathcal{N}\}}\|\mu)$	$\displaystyle\text{ if }\inf\{\mathcal{N}\}>-\infty.$

Definition 1.4.

Let $\mathcal{L}$ denote the space of all measurable functions $f:[0,1]\mapsto{\rm cl}(\mathcal{N})$ such that $\int_{0}^{1}|f(u)|^{p}du<\infty$ . Define a map $\Xi:\mathcal{L}\mapsto\widetilde{\mathcal{M}}$ as follows:

For any $f\in\mathcal{L}$ , if $(U,V)\sim\Xi(f)$ , then $U\sim\mathrm{U}[0,1]$ , and given $U=u$ , one has

$\displaystyle V\sim$	$\displaystyle\mu_{\beta(f(u))}\text{ if }f(u)\in\mathcal{N},$
$\displaystyle=$	$\displaystyle\sup\{\alpha^{\prime}(\mathbb{R})\}\text{ if }f(u)=\sup\{\mathcal{N}\},$	$\displaystyle\text{ (this can only happen if }\sup\{\mathcal{N}\}<\infty),$
$\displaystyle=$	$\displaystyle\inf\{\alpha^{\prime}(\mathbb{R})\}\text{ if }f(u)=\inf\{\mathcal{N}\},$	$\displaystyle\text{ (this can only happen if }\inf\{\mathcal{N}\}>-\infty).$

Definition 1.5.

Fix $W\in\mathcal{W}$ and let $\mathcal{L}$ be as defined above. Define the functional $G_{W}(.):\mathcal{L}\mapsto\mathbb{R}$ by setting

G_{W}(f):=\int_{[0,1]^{v}}\left(\prod_{(a,b)\in E(H)}W(x_{a},x_{b})\right)\left(\prod_{a=1}^{v}f(x_{a})dx_{a}\right),

whenever $G_{|W|}(|f|)<\infty$ (see 1.1 below for sufficient conditions).

Finally, let $\mathfrak{L}_{n}(\cdot)$ be a map from $\mathbb{R}^{n}$ to $\mathcal{M}$ defined by

(1.7)

{\color[rgb]{0,0,0}\mathfrak{L}_{n}(\mathbf{x})}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{(\frac{i}{n},x_{i})},\qquad\mathbf{x}=(x_{1},\ldots,x_{n}).

The following proposition characterizes the asymptotics of the log partition function/free energy in terms of an infinite dimensional optimization problem, and gives a characterization for the class of optimizers in terms of a fixed point equation.

Proposition 1.1.

Suppose that $\mu$ satisfies (1.6) for some $p\geq v$ and all $\lambda>0$ . Let $\{Q_{n}\}_{n\geq 1}$ be a sequence of matrices such that (1.4) holds for some $W\in\mathcal{W}$ , and

(1.8)

\limsup_{n\to\infty}\ \lVert W_{Q_{n}}\rVert_{q\Delta}<\infty,

for some $q>1$ such that $\frac{1}{p}+\frac{1}{q}\leq 1$ . Then the following conclusions hold.

(i) The function $G_{W}(.)$ is well-defined on $\mathcal{L}$ , i.e., $G_{|W|}(|f|)<\infty$ for all $f\in\mathcal{L}$ .

(ii) With $Z_{n}(\theta)$ as in (1.1), we have $\sup_{n\geq 1}Z_{n}(\theta)<\infty$ and

(1.9)		$\displaystyle\lim_{n\to\infty}Z_{n}(\theta)$	$\displaystyle=$	$\displaystyle\sup_{t\in\mathbb{R}:I(t)<\infty}\{\theta t-I(t)\}$
(1.9)			$\displaystyle=$	$\displaystyle\sup_{f\in\mathcal{L}:\ \int_{[0,1]}\gamma(\beta(f(x)))dx<\infty}\left\{\theta G_{W}(f)-\int_{[0,1]}\gamma(\beta(f(x)))dx\right\}=:Z(\theta).$

(iii) The supremum in (1.9) is achieved on a set $F_{\theta}\subset\mathcal{L}$ (say), which satisfies

(1.10)

d_{\ell}(\mathfrak{L}_{n}({\bf X}),\Xi(F_{\theta}))\overset{P}{\longrightarrow}0

under ${\bf X}\sim\mathbb{R}_{n,\theta}$ (as in (1.3)), where $\Xi$ is defined by 1.4. Further $\Xi(F_{\theta})$ is compact in the weak topology.

The above proposition follows from [4, Theorems 1.1 and 1.6].

Remark 1.1.

Under assumptions (1.4) and (1.8), [9, Theorem 2.13] gives

(1.11)

\|W\|_{q\Delta}<\infty,

for any $q>1$ , a fact that we use throughout the paper. We note in passing that under stronger assumptions on $H$ and $\mu$ (similar to [4, Theorem 1.2]) it is possible to forego the requirement in (1.8) and replace it with weaker assumptions.

1.1.1. Replica-symmetry

The above proposition shows that the infinite dimensional optimization problem in the second line of (1.9) is useful for understanding the Gibbs measure $\mathbb{R}_{n,\theta}$ . (see parts (iii) and (iv)). A natural question is when does the set of optimizers of (1.9) consist only of constant functions. Equivalently, borrowing terminology from statistical physics, we want to understand the “replica-symmetry” phase of the Gibbs measure $\mathbb{R}_{n,\theta}$ . Our first main result provides necessary and sufficient conditions for optimizers to be constant functions. For this we need the following two definitions.

Definition 1.6.

Given a symmetric matrix $Q_{n}$ , define a symmetric tensor

\mathrm{Sym}[Q_{n}](i_{1},\ldots,i_{v}):=\frac{1}{v!}\sum_{\sigma\in S_{v}}\prod_{(a,b)\in E(H)}Q_{n}\Big{(}i_{\sigma(a)},i_{\sigma(b)}\Big{)},

where $S_{v}$ denote the set of all permutations of $[v]$ . In a similar vein, given a symmetric function $W\in\mathcal{W}$ , define the symmetric function

\mathrm{Sym}[W](x_{1},\ldots,x_{v}):=\frac{1}{v!}\sum_{\sigma\in S_{v}}\prod_{(a,b)\in E(H)}W\Big{(}x_{\sigma(a)},x_{\sigma(b)}\Big{)}.

As an example, if $H=K_{1,2}$ , $\mathrm{Sym}[W](x_{1},x_{2},x_{3})$ equals

\displaystyle\frac{1}{3}\Big{[}W(x_{1},x_{2})W(x_{1},x_{3})+W(x_{1},x_{2})W(x_{2},x_{3})+W(x_{1},x_{3})W(x_{2},x_{3})\Big{]},

whereas if $H=K_{3}$ , $\mathrm{Sym}[W](x_{1},x_{2},x_{3})$ equals $W(x_{1},x_{2})W(x_{1},x_{3})W(x_{2},x_{3}).$

Let

(1.12)

\displaystyle\mathcal{T}[\mathrm{Sym}[W]](x):=\int_{[0,1]^{v-1}}\mathrm{Sym}[W](x,x_{2},\ldots,x_{v})\prod_{a=2}^{v}\,dx_{a},

provided the integral exists and is finite.

Definition 1.7.

Let $\mu$ be a measure in $\mathbb{R}$ and $\beta(.),\gamma(.),\mathcal{N}$ be as in 1.3. We will say $\mu$ is stochastically non-negative, if for any $t>0$ , if $-t\in\mathcal{N}$ then $t\in\mathcal{N}$ , and $\gamma(\beta(t))\leq\gamma(\beta(-t)).$

We now state the first main result of this paper.

Theorem 1.2 (Replica-symmetry).

Suppose we are in the setting of 1.1. Then $\mathcal{T}[\mathrm{Sym}[W]](.)$ is finite a.s., and the following conclusions hold:

(i) Any maximizer $f$ of the optimization problem (1.9) satisfies

(1.13)

f(x)\stackrel{{\scriptstyle a.s.}}{{=}}\alpha^{\prime}\left(\theta v\int_{[0,1]^{v-1}}\mathrm{Sym}[W](x,x_{2},\ldots,x_{v})\left(\prod_{a=2}^{v}f(x_{a})\,dx_{a}\right)\right).

(ii) If $\theta\neq 0$ and $\mathcal{T}[\mathrm{Sym}[W]](\cdot)$ is not constant a.s., none of the maximizers in (1.9) are non-zero constant functions.

(iii) If $\mathcal{T}[\mathrm{Sym}[W]](\cdot)$ is constant a.s., and $\theta W$ is strictly positive a.s., then all of the maximizers in (1.9) are constant functions, provided either $v$ is even or $\mu$ is stochastically non-negative.

(iv) $\mu$ is stochastically non-negative if one of the following conditions hold:

(a) $\mu$ is supported on the non-negative half line, or

(b) $\mu$ is a non-negative tilt of a symmetric measure, i.e. there exists $B\geq 0$ and a symmetric measure $\tilde{\mu}$ , such that $\frac{d\mu}{d\tilde{\mu}}(x)=\exp(Bx-C(x))$ .

Remark 1.2.

It follows from the construction of the map $\Xi$ in 1.4 that if $f\in\mathcal{L}$ is a constant function, then $\Xi(f)\in\widetilde{\mathcal{M}}$ is a product measure. Thus under the conditions of 1.2 part (iii), any weak limit of the empirical measure $\mathfrak{L}_{n}$ (introduced in (1.7)) under $\mathbb{R}_{n,\theta}$ is a product measure. Further these product measures have first marginal $\mathrm{Unif}[0,1]$ and second marginal of the form $\mu_{\beta(t)}$ where $t$ satisfies the fixed point equation

t=\alpha^{\prime}(\theta vt^{v-1}),

by 1.2 part (i). In particular if $v=2$ and $\mu$ is supported on $\{-1,1\}$ with $\mu(1)=\exp(2B)/(1+\exp(2B))$ for some $B\in\mathbb{R}$ , the above fixed point equation simplifies to

t=\tanh(2\theta t+B).

The solutions of this equation for $\theta\geq 0$ , $B\in\mathbb{R}$ are well understood, see e.g., [14, Page 2] and [18, page 144, Section 1.1.3]. In 1.7 below we study a broader class of measures $\mu$ , which includes this as a special case.

Example 1.1 (Necessity of conditions on $\mu$ ).

To demonstrate the necessity of stochastic non-negativeness for $\mu$ in 1.2 part (iii), we provide an example of a measure $\mu$ and a graphon $W$ with $\mathcal{T}[\mathrm{Sym}[W]](\cdot)$ is constant a.s. and $\theta W\geq 0$ , but none of the maximizers are constant functions. Set $H=K_{3}$ and

	$\displaystyle W(x,y)=$	$\displaystyle 0\text{ if }(x,y)\in\Big{[}0,\frac{1}{3}\Big{)}^{2}\text{ or }(x,y)\in\Big{[}\frac{1}{3},\frac{2}{3}\Big{)}^{2}\text{ or }(x,y)\in\Big{[}\frac{2}{3},1\Big{)}^{2},$
	$\displaystyle=$	$\displaystyle 1\text{ otherwise }.$

In this case $W$ is the graphon corresponding to a complete tripartite graph. Let $\mu$ be a probability measure on $\{-1,1\}$ with $\mu(1)=e^{-4}/(e^{-4}+e^{4})$ . Set $\theta=9$ and

\displaystyle f(x):=\begin{cases}-0.99&\text{ if }0\leq x<\frac{2}{3}\\ +0.83&\text{ if }\frac{2}{3}\leq x\leq 1.\end{cases}

Numerical computations show that

\displaystyle\theta G_{W}(f)-\int_{[0,1]}\gamma(\beta(f(x)))dx>\sup_{t\in[-1,1]}\left\{\frac{2}{3}\theta t^{v}-\gamma(\beta(t))\right\}.

Thus all global optimizers must be non-constant functions when $v=3$ , and the measure $\mu$ is a negative tilt of a symmetric distribution. Note that the function $W$ in this counterexample is not strictly positive a.s., but this can be circumvented by a continuity argument to allow for small positive values in the diagonal blocks.

Example 1.2 (Necessity of $\theta>0$ ).

The requirement $\theta>0$ is indeed necessary in 1.2 part (iii). To see a counterexample, consider the case $v=2$ and

	$\displaystyle W(x,y)=$	$\displaystyle 2\text{ if }(x,y)\in(0,.5)\times(.5\times 1)\text{ or }(x,y)\in(.5,1)\times(0,.5),$
	$\displaystyle=$	$\displaystyle 0\text{ otherwise }.$

Let $\mu$ be a compactly supported probability measure which is symmetric about $0$ . If $\theta$ is large and negative, then one can show numerically that no optimizer of (1.9) is a constant function, and any optimizer is of the form

	$\displaystyle f(x)=$	$\displaystyle a\text{ if }0<x<0.5;$
	$\displaystyle=$	$\displaystyle b\text{ if }.5<x<1,$

where $a$ and $b$ are of opposite signs. Similar to the previous remark, even though $W$ is not strictly positive a.s., this can be circumvented by a continuity argument to allow for small positive values in the diagonal blocks.

1.1.2. Weak laws and tail bounds

Having studied the optimizers of the limiting free energy under model (1.3) in 1.2, the next natural question is to obtain weak laws for various statistics of interest under (1.3). Some popular examples include the Hamiltonian $\mathbb{U}_{n}$ (see (1.2)), the global magnetization $\sum_{i=1}^{n}X_{i}$ or other interesting linear combinations of $X_{i}$ ’s. In the sequel, we will obtain a family of such weak limits in a unified fashion. Along the way, we will provide a probabilistic interpretation of the optimizers of the limiting free energy. The key tool that will help us address these question simultaneously is a sharp analysis of the vector of local fields, which we define below.

Definition 1.8 (Local fields).

Define the local magnetization/field at the $i$ -th observation as follows:

(1.14)

\displaystyle m_{i}:=\frac{v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in\mathcal{S}(n,v,i)}\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\left(\prod_{a=2}^{v}X_{i_{a}}\right),

for $i\in[n]$ . Here, for $i\in[n]$ , $\mathcal{S}(n,v,i)$ denotes the set of all distinct tuples in $[n]^{v-1}$ , such that none of the elements equal to $i$ , and $\mathrm{Sym}[Q_{n}]$ is as in 1.6. Set $\mathbf{m}:=(m_{1},\ldots,m_{n})$ . Following (1.7), consider the associated empirical measure

(1.15)

\mathfrak{L}_{n}(\mathbf{m})=\frac{1}{n}\sum_{i=1}^{n}\delta_{\left(\frac{i}{n},m_{i}\right)},

The $m_{i}$ ’s defined in (1.14) are local magnetizations/fields which capture “how well” one can predict $X_{i}$ given all $X_{j}$ , $j\neq i$ . More precisely, under model (1.3), the conditional distribution of $X_{i}$ given $X_{j}$ , $j\neq i$ is completely determined by $m_{i}$ in the following manner:

(1.16)

\frac{d\mathbb{R}_{n,\theta}(x_{i}|x_{j},\ j\neq i)}{d\mu(x_{i})}=\mu_{\theta m_{i}},

with $\mu_{.}$ as in 1.3. In particular, by (1.16) and 1.3,

(1.17)

\mathbb{E}[X_{i}|X_{j},\ j\neq i]=\alpha^{\prime}(\theta m_{i}).

Consequently, understanding the behavior of the vector ${\bf m}$ or

\boldsymbol{\alpha}:=\alpha^{\prime}(\theta{\bf m})=(\alpha^{\prime}(\theta m_{1}),\ldots,\alpha^{\prime}(\theta m_{n})),

plays an important role in obtaining correlation bounds, tail decay estimates and fluctuations for such Gibbs measures (see [22, 13, 17, 16, 7] and the references therein). The major focus of the existing literature is on the special case of $v=2$ and $\mu$ supported on $\{-1,1\}$ , which is not needed in our paper.

Our second main result gives a weak law for the vector ${\bf m}$ , in terms of the empirical measure $\mathfrak{L}_{n}(\mathbf{m})$ . For any measure $\nu\in\widetilde{\mathcal{M}}_{p}$ (see (1.5)), define

(1.18)

\vartheta_{W,\nu}(u):=v\mathbb{E}\left[\mathrm{Sym}[W](U_{1},\ldots,U_{v})\left(\prod_{a=2}^{v}V_{a}\right)\bigg{|}U_{1}=u\right],\quad\mbox{for}\ u\in[0,1],

where $(U_{1},V_{1}),\ldots,(U_{v},V_{v})\overset{i.i.d.}{\sim}\nu$ , and $\mathrm{Sym}[.]$ is as in 1.6. In the Theorem below, we will provide sufficient conditions under which $\vartheta_{W,\nu}(\cdot)$ is well-defined. While the definition of $\vartheta_{W,\nu}$ may seem abstract at first, it simplifies nicely in the context of 1.2. To see this, assume that $(U,V)\sim\nu\in\Xi(F_{\theta})$ for some $\theta\in\mathbb{R}$ , where $F_{\theta}$ is as in 1.1 part (iv). By 1.4 we have $\mathbb{E}[V|U=u]=f(u)$ for some $f\in F_{\theta}$ . So,

	$\displaystyle\vartheta_{W,\nu}(u)$	$\displaystyle=v\mathbb{E}\left[\mathrm{Sym}[W](u,U_{2},\ldots,U_{v})\left(\prod_{a=2}^{v}\mathbb{E}[V_{a}\|U_{a}]\right)\right]$
		$\displaystyle=v\int_{[0,1]^{v-1}}\mathrm{Sym}[W](u,u_{2},\ldots,u_{v})\left(\prod_{a=2}^{v}f(u_{a})\,du_{a}\right).$

By invoking (1.13), we then get for a.e. $u\in[0,1]$ ,

(1.19)

f(u)=\alpha^{\prime}(\theta\vartheta_{W,\nu}(u)).

Therefore, there is a direct one-one correspondence between the two sets of functions $F_{\theta}$ and $\{\vartheta_{W,\nu}(.),\nu\in\Xi(F_{\theta})\}$ . This observation will be crucial in the proof of 1.5 below.

We are now in a position to state our second main result.

Theorem 1.3.

Suppose (1.6) holds for some $p\in[v,\infty]$ . Assume that (1.4) and (1.8) holds with some $q$ satisfying $\frac{1}{p}+\frac{1}{q}<1$ . Then the following conclusions hold:

(i) With $F_{\theta}$ as in 1.1 part (iv), and $\vartheta_{W,\nu}$ as in (1.18), we have $\Xi(F_{\theta})\subseteq\widetilde{\mathcal{M}}_{p}$ and $\vartheta_{W,\nu}(\cdot)$ is well-defined for every $\nu\in\widetilde{\mathcal{M}}_{p}$ , a.s. on $[0,1]$ .

(ii) Set

\mathfrak{B}^{*}_{\theta}:=\{\mathrm{Law}(U,\vartheta_{W,\nu}(U)):\nu\in\Xi(F_{\theta})\}\subset\widetilde{\mathcal{M}}.

Then we have:

(1.20)

d_{\ell}\left(\mathfrak{L}_{n}(\mathbf{m}),\mathfrak{B}^{*}_{\theta}\right)\overset{P}{\longrightarrow}0.

The above theorem gives a weak limit for the empirical measure of the local field vector $\bf m$ . The weak law for the empirical measure of the conditional means (introduced in (1.17)) then follows from 1.3 by a continuous mapping type argument. The limit in that case will naturally be the set

(1.21)

\displaystyle\widetilde{\mathfrak{B}}_{\theta}:=\{\mbox{Law}(U,\alpha^{\prime}(\theta\vartheta_{W,\nu}(U))):\ \nu\in\Xi(F_{\theta})\},

We stress here that no assumption of replica-symmetry is necessary for 1.3 to hold.

Remark 1.3.

Given $\nu\in\Xi(F_{\theta})$ , let $\nu^{(2|1)}(\cdot)$ denote the conditional distribution of the second coordinate given the first coordinate. The proof of (1.20) can be adapted to show the following stronger conclusion:

d_{\ell}\left(\frac{1}{n}\sum_{i=1}^{n}\delta_{\left(\frac{i}{n},X_{i},m_{i}\right)},\underline{\mathfrak{B}}_{\theta}\right)\overset{P}{\longrightarrow}0,

where

\underline{\mathfrak{B}}_{\theta}:=\{(U,\nu^{(2|1)}(U),\vartheta_{W,\nu}(U)):\ \nu\in\Xi(F_{\theta})\}.

Since this version is not necessary for our applications, we do not prove it here.

In order to obtain weak laws for common statistics of interest using 1.3, we require appropriate tail estimates for the $X_{i}$ ’s, the $m_{i}$ ’s and the $\alpha^{\prime}(\theta m_{i})$ ’s. In particular, we will derive exponential tail bounds for the said quantities below, which is of possible independent interest.

Theorem 1.4.

Consider the same setting as in 1.1. Then the following conclusions hold:

(i)

There exists $C_{0}>0$ , free of $n$ , such that for all $C\geq C_{0}$ we have

(1.22)	$\displaystyle\limsup\limits_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}\|X_{i}\|^{p}\geq C\right)<0$	$\displaystyle\Rightarrow\sum_{i=1}^{n}\mathbb{E}\|X_{i}\|^{p}=O(n),$
(1.23)	$\displaystyle\limsup\limits_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|^{q}\geq C\right)<0$	$\displaystyle\Rightarrow\sum_{i=1}^{n}\mathbb{E}\|m_{i}\|^{q}=O(n),$
(1.24)	$\displaystyle\limsup\limits_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})\|^{p}\geq C\right)<0$	$\displaystyle\Rightarrow\sum_{i=1}^{n}\mathbb{E}\|\alpha^{\prime}(\theta m_{i})\|^{p}=O(n).$

(ii)

Moreover,

(1.25)

\displaystyle\sup_{\nu\in\Xi(F_{\theta})}\mathfrak{m}_{p}(\nu)<\infty,\quad\sup_{\nu\in\mathfrak{B}_{\theta}^{*}}\mathfrak{m}_{q}(\nu)<\infty,\quad\sup_{\nu\in\widetilde{\mathfrak{B}}_{\theta}}\mathfrak{m}_{p}(\nu)<\infty,

where $\mathfrak{m}_{p}(\nu),\mathfrak{m}_{q}(\nu)$ are given by 1.2.

In order to interpret part (ii) of the above theorem, note that $\mathfrak{L}_{n}({\bf X})$ , $\mathfrak{L}_{n}({\bf m})$ , and $\mathfrak{L}_{n}({\boldsymbol{\alpha}})$ converge weakly in probability to the set of probability measures $\Xi(F_{\theta})$ (by (1.10)), $\mathfrak{B}_{\theta}^{*}$ (by (1.20)), and $\widetilde{\mathfrak{B}}_{\theta}$ (as discussed around (1.21)) respectively. 1.4 part (ii) shows that these limiting set of measures have uniformly bounded moments of a suitable order.

In view of the above results, coupled with the observation made in (1.19), it seems intuitive to expect a correspondence between elements of $F_{\theta}$ and the map $u\mapsto\alpha^{\prime}(\theta m_{\lceil nu\rceil}))$ . This is made precise in the following corollary.

Corollary 1.5.

In the setting of 1.3, we have

(1.26)

\inf_{f\in F_{\theta}}\int_{0}^{1}|\alpha^{\prime}(\theta m_{\lceil nu\rceil})-f(u)|^{p^{\prime}}\,du\overset{P}{\longrightarrow}0,

for any $p^{\prime}<p$ , under the measure (1.3).

Remark 1.4.

Recall from (1.17) that $\alpha^{\prime}(\theta m_{i})=\mathbb{E}[X_{i}|X_{j},\ j\neq i]$ . Therefore (1.26) shows that the functions in $F_{\theta}$ are “close” to the vector of conditional expectations of $X_{i}$ ’s given all the other coordinates. In particular, if $F_{\theta}=\{f\}$ is a singleton and $f$ is continuous on $[0,1]$ , then (1.26) and (1.17) together imply that

\frac{1}{n}\sum_{i=1}^{n}\bigg{|}\alpha^{\prime}(\theta m_{i})-f\left(\frac{i}{n}\right)\bigg{|}^{p^{\prime}}\overset{P}{\longrightarrow}0.

For the special case where $v=2$ , $\mu$ is supported on $\{-1,1\}$ with $\mu(1)=\mu(-1)=0.5$ , and $\mathcal{T}[\mathrm{Sym}[W]](\cdot)=1$ a.s, $F_{\theta}$ consists of two constant functions of the form $\{-t_{\theta},t_{\theta}\}$ for some $t_{\theta}>0$ (see [18, Page 144]; also see 1.7 for general $\mu$ ). The symmetry of $\mathbb{R}_{n,\theta}$ around $0$ , coupled with (1.26) implies

\frac{1}{n}\sum_{i=1}^{n}\bigg{|}|\tanh(\theta m_{i})|-t_{\theta}\bigg{|}\overset{P}{\longrightarrow}0.

Remark 1.5.

Note that the above displays are not true with $X_{i}$ replacing $\tanh(\theta m_{i})=\mathbb{E}[X_{i}|X_{j},\ j\neq i]$ . This shows $m_{i}$ ’s are “more concentrated” than $X_{i}$ ’s.

As applications of 1.3 and 1.4, we obtain weak limits for linear statistics as well as the Hamiltonian $\mathbb{R}_{n,\theta}$ under replica-symmetry, both of which are of independent interest.

Theorem 1.6.

Suppose ${\mathbf{X}}\sim\mathbb{R}_{n,\theta}$ (defined via (1.3)) for some base measure $\mu$ which satisfies (1.6) for some $p\in[v,\infty]$ . Assume that either $v$ is even or $\mu$ is stochastically non-negative. Further, suppose that $\{Q_{n}\}_{n\geq 1}$ satisfies (1.4) and (1.8) for some $q>1$ satisfying $\frac{1}{p}+\frac{1}{q}<1$ , and that the limiting $W$ is strictly positive and satisfies $\mathcal{T}[\mathrm{Sym}[W]](x)=1$ a.s. Then, setting

(1.27)

\displaystyle\mathcal{A}_{\theta}:=\operatorname{argmin}_{t\in\mathcal{N}}\left[\gamma(\beta(t))-\theta t^{v}\right],

we have $\mathcal{A}_{\theta}$ is a finite set, and the following conclusions hold:

(i) Suppose $\{c_{i}\}_{i\geq 1}$ is a real sequence satisfying $\sum_{i=1}^{n}c_{i}=o(n)$ , and $\sum_{i=1}^{n}|c_{i}|^{r}=O(n)$ for some $r$ such that $\frac{1}{p}+\frac{1}{r}<1$ . Then we have:

\frac{1}{n}\sum_{i=1}^{n}c_{i}X_{i}\overset{P}{\longrightarrow}0.

(ii) If we replace $\sum_{i=1}^{n}c_{i}=o(n)$ with $n^{-1}\sum_{i=1}^{n}c_{i}\to\tilde{c}$ , then

d_{\ell}\left(\frac{1}{n}\sum_{i=1}^{n}c_{i}X_{i},\{{\tilde{c}t}:\ t\in\mathcal{A}_{\theta}\}\right)\overset{P}{\longrightarrow}0.

(iii) The Hamiltonian satisfies

d_{\ell}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}m_{i},\{{vt^{v}}:\ t\in\mathcal{A}_{\theta}\}\right)\overset{P}{\longrightarrow}0.

Remark 1.6.

Part (i) of the above Theorem shows that for contrast vectors ${\bf c}$ (i.e. $\sum_{i=1}^{n}c_{i}=0$ ) which are delocalized (in the sense $\sum_{i=1}^{n}|c_{i}|^{r}=O(n)$ ), the corresponding linear statistic exhibits a universal behavior across general Gibbs measures with higher order multilinear interactions which doesn’t depend on the matrix sequence $\{Q_{n}\}_{n\geq 1}$ , as long as $\mathcal{T}[\mathrm{Sym}[W]](.)$ is constant, i.e. the symmetrized tensor is regular. In a similar manner, part (ii) gives a universal behavior for the global magnetization $\bar{X}$ for regular tensors. Universality results for $\bar{X}$ were earlier obtained for regular Ising models, which correspond to $v=2$ and $\mu$ is supported on $\{-1,1\}$ with $\mu(-1)=\mu(+1)=0.5$ (see [3, Theorem 2.1] and [17, Theorems 1.1—1.4]). In this special case, for $\theta>0.5$ we have $\mathcal{A}_{\theta}=\{-t_{\theta},t_{\theta}\}$ for some $t_{\theta}>0$ (see Remark 1.4). In this case symmetry implies (see 1.8 part (ii) below for a more general result) that:

\bar{X}\overset{d}{\longrightarrow}\frac{\delta_{-t_{\theta}}+\delta_{t_{\theta}}}{2},\quad\frac{1}{n}\sum_{i=1}^{n}X_{i}m_{i}\overset{P}{\longrightarrow}2t_{\theta}^{2}.

The more recent work of [25] demonstrates universality for quadratic interactions for log concave $\mu$ (see [25, Theorem 1.1 and Corollary 1.4]). We note that the results of the current paper requires neither quadratic interactions, nor log-concave base measures. In the following subsection, we will apply 1.6 to analyze a broad class of Gibbs measures which are not necessarily quadratic, and cover cubic and higher order interactions (see 1.9 below).

1.2. Examples

We now apply our general results to analyze some Gibbs measures of interest. In 1.2, we proved that for regular tensors (i.e. when $\mathcal{T}[\mathrm{Sym}[W]](\cdot)=1$ a.s.) the optimization problem (1.9) has only constant functions as optimizers, under mild assumptions on $\mu$ or $v$ . In this section, we focus on particular examples of the regular case, and provide more explicit description for the optimizers.

1.2.1. Quadratic interaction models with symmetric base measure

Suppose $\mu$ is a probability measure on $\mathbb{R}$ which is symmetric about the origin. Define a Gibbs measure on $\mathbb{R}^{n}$ by setting

(1.28)

\displaystyle\frac{d\mathbb{R}^{\mathrm{quad}}_{n,\theta,B}}{d\mu^{\otimes n}}({\bf X})=\exp\Big{(}\frac{\theta}{n}\sum_{i\neq j}Q_{n}(i,j)X_{i}X_{j}+B\sum_{i=1}^{n}X_{i}-nZ^{\mathrm{quad}}_{n}(\theta,B)\Big{)},

where $\theta\geq 0$ , $B\in\mathbb{R}$ . In particular if $\mu$ is supported on $\{-1,1\}$ reduces the above model to the celebrated Ising model, which has attracted significant attention in probability and statistics (c.f. [3, 18, 20, 30, 17] and references there-in). The following results analyzes the optimization problem (1.9) in the particular setting (which corresponds to setting $H=K_{2}$ ).

Lemma 1.7.

Let $\mu$ be a probability measure symmetric about $0$ , which satisfies (1.6) with $p\geq 2$ , and let $\alpha(\cdot)$ , $\beta(\cdot)$ , and $\mathcal{N}$ be as in 1.3. Assume that

(1.29)

\alpha^{\prime\prime}(x)\leq\alpha^{\prime\prime}(y)\quad\mbox{ for all }\quad|x|\geq|y|.

Then, setting

\mathfrak{v}_{\theta,B,\mu}(x):=\theta x^{2}+Bx-x\beta(x)+\alpha(\beta(x))

for $\theta\geq 0$ , $B\in\mathbb{R},x\in{\rm cl}(\mathcal{N})$ , the following conclusions hold:

(i)

If $2\theta\leq(\alpha^{\prime\prime}(0))^{-1}$ and $B=0$ , then $\mathfrak{v}_{\theta,B,\mu}(\cdot)$ has a unique maximizer at $t_{\theta,B,\mu}=0$ .
(ii)

If $B\neq 0$ , then $\mathfrak{v}_{\theta,B,\mu}(\cdot)$ has a unique maximizer $t_{\theta,B,\mu}$ with the same sign as that of $B$ which satisfies $t_{\theta,B,\mu}=\alpha^{\prime}(2\theta t_{\theta,B,\mu}+B)$ .
(iii)

If $2\theta>(\alpha^{\prime\prime}(0))^{-1}$ and $B=0$ , then $\mathfrak{v}_{\theta,B,\mu}(\cdot)$ has two maximizers $\pm t_{\theta,B,\mu}$ , where $t_{\theta,B,\mu}>0$ , and $t_{\theta,B,\mu}=\alpha^{\prime}(2\theta t_{\theta,B,\mu})$ .

The proof of 1.7 is provided in Section 3.

Remark 1.7.

A few comments about the extra condition (1.29) utilized in the above lemma are in order. First note that that if any measure $\mu$ satisfies the celebrated GHS inequality of statistical physics (see [19]), then $\mu$ must satisfy (1.29). Indeed, taking the matrix $J$ in [19, (1.2)] to be the ${\bf 0}$ matrix, it follows on applying the GHS inequality ([19, (1.4)] that $\alpha^{\prime\prime\prime}(\theta)\leq 0$ for $\theta\geq 0$ , which immediately implies (1.29). Sufficient conditions on $\mu$ for the GHS inequality (and hence (1.29)) can be found in [19, Theorem 1.2]. In [19, (1.5)] the authors give a counterexample where GHS inequality fails. Using the same example, it is not hard to show that (1.29) fails in this case as well, and $\mathfrak{v}(\cdot)$ does not have a unique maximizer for $B=0$ and some $\theta\leq(\alpha^{\prime\prime}(0))^{-1}/2$ .

Proposition 1.8.

Suppose that the measure $\mu$ satisfies the assumptions of 1.7, and $\{Q_{n}\}_{n\geq 1}$ satisfy (1.4) and (1.8) with $H=K_{2}$ (i.e. $\Delta=1$ ). Also assume that the limiting graphon $W$ is strictly positive a.s., and satisfies $\int_{0}^{1}W(.,y)dy=1$ a.s. With $\mathcal{L}$ as in 1.4, define the functional $G_{W}:\mathcal{L}\mapsto\mathbb{R}$ by setting

(1.30)

\displaystyle G_{W}(f):=\int_{[0,1]^{2}}W(x,y)f(x)f(y)dxdy.

Then $G_{W}$ is well defined by 1.1 part (i), and further, the following conclusions hold:

(i) For any $\theta\geq 0$ , $B\in\mathbb{R}$ the optimization problem

(1.31)

\displaystyle\sup_{f\in\mathcal{L}}\{\theta G_{W}(f)-\int_{[0,1]}\gamma(\beta(f(x)))dx\}

has only constant global maximizers, given by

{F}_{\theta,B}\equiv\begin{cases}0&\text{ if }\theta\leq(\alpha^{\prime\prime}(0))^{-1}/{2},B=0,\\ \pm{t_{\theta,B,\mu}}&\text{ if }\theta>(\alpha^{\prime\prime}(0))^{-1}/{2},B=0,\\ {t_{\theta,B,\mu}}&\text{ if }B\neq 0\end{cases}

Here $t_{\theta,B,\mu}$ is as in 1.7.

(ii) The following weak limits hold:

\frac{1}{n}\sum_{i=1}^{n}X_{i}m_{i}\overset{d}{\longrightarrow}2t_{\theta,B,\mu}^{2},\qquad\bar{X}\overset{d}{\longrightarrow}\begin{cases}0&\text{ if }\theta\leq(\alpha^{\prime\prime}(0))^{-1}/{2},B=0,\\ \frac{\delta_{t_{\theta,B,\mu}}+\delta_{-t_{\theta,B,\mu}}}{2}&\text{ if }\theta>(\alpha^{\prime\prime}(0))^{-1}/{2},B=0,\\ {t_{\theta,B,\mu}}&\text{ if }B\neq 0\end{cases}.

Remark 1.8.

We note here that in contrast to 1.6 part (iii) which only allows us to identify possible limit points for the random variable $\frac{1}{n}\sum_{i=1}^{n}X_{i}m_{i}$ , in this case we are able to identify the limit, even in the low temperature regime $\theta>(\alpha^{\prime\prime}(0))^{-1}/2$ . This is because, even though $\mathfrak{v}_{\theta,B,\mu}(.)$ has two roots which are both global optimzers (i.e. in $\mathcal{A}_{\theta}$ defined in 1.6), the optimizers are symmetric, and the limit of the Hamiltonian is the same under both optimizers.

1.2.2. Gibbs measures with higher order interactions

We now focus on Gibbs measures with higher order interactions, which has gained significant attention in recent years (see [29, 2, 23, 26, 31, 32, 33, 6] and the references therein). Here, we analyze the optimization problem (1.27), under some conditions on $\theta$ and $\mu$ . We point the reader to [5, Section 2.1] for related results in the special case where $\mu$ is supported on $\{-1,1\}$ .

Theorem 1.9.

Consider the optimization problem

(1.32)

\displaystyle\sup_{f\in\mathcal{L}:\ \int_{[0,1]}\gamma(\beta(f(x)))dx<\infty}\left\{\theta G_{W}(f)+B\int f(x)dx-\int_{[0,1]}\gamma(\beta(f(x)))dx\right\}.

Then the following conclusions hold:

(i) Fixing $\theta\in\mathbb{R}$ , the maximizers of the optimization problem are attained and satisfies the equation

(1.33)

\displaystyle f(x)\stackrel{{\scriptstyle a.s.}}{{=}}\alpha^{\prime}\left(\theta v\int_{[0,1]^{v-1}}\mathrm{Sym}[W](x,x_{2},\ldots,x_{v})\left(\prod_{a=2}^{v}f(x_{a})\,dx_{a}\right)+B\right).

(ii) If $\mathcal{T}[\mathrm{Sym}[W]](\cdot)$ is constant a.s., and $W$ is strictly positive a.s., and $\theta,B\geq 0$ , then all of the maximizers are constant functions, provided provided either $v$ is even or $\mu$ is stochastically non-negative. Further any such constant maximizer $x$ satisfies

(1.34)

x\stackrel{{\scriptstyle a.s.}}{{=}}\alpha^{\prime}\left(\theta vx^{v-1}+B\right).

(iii) Suppose further that $\mu$ is compactly supported on $[-1,1]$ Then the following hold:

(a)

There exists $B_{0}=B_{0}(\theta,v)$ such that if $B>B_{0}$ , the optimization problem has a unique maximizer.
(b)

If $B=0$ and $\alpha^{\prime}(0)=0$ , there exists $\theta_{c}\in(0,\infty)$ such that if $\theta<\theta_{c}$ , the optimization problem has the unique maximizer $x=0$ , whereas if $\theta>\theta_{c}$ , then $x=0$ is not a global maximizer.

1.3. Proof overview and future scope

Let us discuss the proof techniques employed in the characterization of replica symmetry (see 1.2) and weak laws (see Theorems 1.3 and 1.6). In 1.2 part (i), we establish the first-order conditions (in (1.13)) for the optimization problem (1.9). It is immediate from (1.13) that if all optimizers of (1.9) are constants, then $\mathcal{T}[\mathrm{Sym}[W]](\cdot)$ is a constant function (1.2, part (ii)). For the other direction, if $\mathcal{T}[\mathrm{Sym}[W]](\cdot)$ is a constant, the crucial observation is that $\mathrm{Sym}[W](x_{1},\ldots,x_{v})$ is a (possibility un-normalized) probability density function on $[0,1]^{v}$ , with $\mathrm{Unif}[0,1]$ marginals. The conclusion in 1.2 part (iii) then follows from the equality conditions of Hölder’s inequality. We provide examples to demonstrate that our conditions required for replica symmetry are essentially tight.

The weak limits we prove involve a number of technical steps. We distil some of the main ideas here in the context of the universal result that $n^{-1}\sum_{i=1}^{n}c_{i}X_{i}\overset{P}{\longrightarrow}0$ provided $\sum_{i=1}^{n}c_{i}=o(n)$ , under any multilinear Gibbs in the replica symmetry phase (see 1.6 part (ii)). For simplicity, let us assume that the optimization problem (1.9) has a unique constant optimizer, say $f(x)\equiv t$ (note that the actual result does not require uniqueness of optimizers). Let us split the proof outline into a few steps.

Step (i). Recall the definition of $m_{i}$ from (1.14). We first show that

\frac{1}{n}\sum_{i=1}^{n}c_{i}(X_{i}-\mathbb{E}[X_{i}|X_{j},\ j\neq i])=\frac{1}{n}\sum_{i=1}^{n}c_{i}(X_{i}-\alpha^{\prime}(\theta m_{i}))=o_{P}(1).

This is the subject of 2.10 part (a), and proceeds with a second moment argument, after a suitable truncation. The above display now suggests that it is sufficient to show that $n^{-1}\sum_{i=1}^{n}c_{i}\alpha^{\prime}(\theta m_{i})\overset{P}{\longrightarrow}0$ .

Step (ii). Based on step (i), it is natural to focus on the vector of local fields ${\bf m}=(m_{1},\ldots,m_{n})$ . The advantage of working with ${\bf m}$ rather than ${\bf X}$ is that each $m_{i}$ is a $(v-1)$ -th order “weighted average”, and hence they are much more “concentrated” than $X_{i}$ ’s. We provide a formalization in 1.3 where (in the current setting) we show that

(1.35)

\frac{1}{n}\sum_{i=1}^{n}\delta_{m_{i}}\overset{d}{\longrightarrow}\delta_{vt^{v-1}},

which is a degenerate limit. In contrast, by 1.1 part (iii), $\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}\overset{d}{\longrightarrow}\mu_{\beta(t)}$ which is a non-degenerate limit. The proof of 1.3 relies primarily on 2.5, which is a stability lemma, the proof of which proceeds relies on counting lemma for $L^{p}$ graphons. In fact, 2.5 can be viewed as a refinement of the counting lemma in [8, Proposition 2.19] for “star-like” graphs.

Step (iii). Based on (1.35) in step (ii), it is natural to consider the following approximation:

\frac{1}{n}\sum_{i=1}^{n}c_{i}\alpha^{\prime}(\theta m_{i})\approx\alpha^{\prime}(\theta vt^{v-1})\frac{1}{n}\sum_{i=1}^{n}c_{i}=o(1).

In the first approximation, we have essentially replaced the $m_{i}$ ’s by the corresponding weak limit from (1.35). To make this rigorous, we need some moment estimates which are immediate byproducts of the exponential tail bounds in 1.4. The final conclusion in the above display uses the condition $\sum_{i=1}^{n}c_{i}=o(n)$ .

More generally, the weak limit of ${\bf m}$ in 1.3 has broad applications. We use it in 1.5 to provide a probabilistic interpretation of the optimizers of (1.9) (note that this does not require the optimizers to be constant functions). We also use 1.3 to derive other weak laws of interest in 1.6. We note in passing that many other statistics of interest, such as the maximum likelihood or the pseudo-maximum likelihood estimators for the inverse temperature parameter are also expressible (sometimes implicitly) as functions of ${\bf m}$ . Consequently one can derive appropriate weak laws for these estimators using 1.3 as well.

Our work leads to several important future research directions. Our results apply, as a special case, to Ising models with quadratic Hamiltonians, and a general base measure. A first question is to extend the techniques of this paper to cover more general Hamiltonians from statistical physics, such as Potts models. Another related question is to go beyond the setting of cut metric convergence, and allow for the matrix $\{Q_{n}\}_{n\geq 1}$ to converge in other topologies (such as local weak convergence on bounded degree graphs). A third question is to study Gibbs measure under more general tensor Hamiltonians, which cannot be specified by a matrix $Q_{n}$ . This would require significant development of cut metric theory for cubic and higher order functions. Finally, it remains to be seen whether we can answer more delicate questions about such Gibbs measures, which include Central Limit Theorems/limit distributions.

1.4. Outline of the paper

In Sections 2 and 3, we prove our main results from Sections 1.1 and 1.2 respectively. The proofs of the major technical lemmas (in the order in which they are presented in the paper) are provided in Section 4. In the Appendix 5, we defer the proof some of our supporting lemmas, which deal with properties of the base measure $\mu$ , and general results on weak convergence.

2. Proof of Main Results

2.1. Proofs of 1.1 and 1.2

In order to prove 1.1, we need the following preparatory result.

Proposition 2.1.

Fix any $v\geq 2$ , $p\geq 1$ , $q>1$ such that $\frac{1}{p}+\frac{1}{q}\leq 1$ and $W\in\mathcal{W}$ . Fix any probability measure $\nu$ supported on $[0,1]\times\mathbb{R}$ with first marginal $\mathrm{Unif}[0,1]$ and sample $(U_{1},V_{1}),\ldots,(U_{v},V_{v})\overset{i.i.d.}{\sim}\nu$ . Then the following conclusions hold:

(i)

We have:

$\displaystyle\mathbb{E}\left(\prod_{(a,b)\in E(H)}|W(U_{a},U_{b})|\right)\leq\|W\|_{\Delta}^{|E(H)|}.$

(ii)

For any measurable $\phi:\mathbb{R}^{v}\to\mathbb{R}$ we have

\displaystyle\mathbb{E}\left[\left(\prod_{(a,b)\in E(H)}|W(U_{a},U_{b})|\right)|\phi(V_{1},\ldots,V_{v})|\right]\leq\|W\|_{q\Delta}^{|E(H)|}\Big{(}\mathbb{E}|\phi(V_{1},\ldots,V_{v})|^{p}\Big{)}^{\frac{1}{p}}.

(iii)

With $\mathrm{Sym}[.]$ as in 1.6, we have

$\displaystyle\mathbb{E}\Big{[}\mathrm{Sym}[|W|](U_{1},\ldots,U_{v})^{q}\Big{]}\leq\|W\|_{q\Delta}^{q|E(H)|}.$

Parts (i) and (ii) above follow from [10, Proposition 2.19] and [6, Lemma 2.2] respectively. However, part (iii) is new and a proof is provided in Section 5.2. While the proof of 1.1 only uses 2.1 part (ii), the other parts of 2.1 will be useful in the rest of the paper.

Remark 2.1.

When the RHS of the display in part (ii) of 2.1 is finite, we can define

\displaystyle T_{W,\phi}(\nu):=\mathbb{E}\left[\left(\prod_{(a,b)\in E(H)}W(U_{a},U_{b})\right)\phi(V_{1},\ldots,V_{v})\right].

Proof of 1.1.

Under the conditions of 1.1, 2.1 part (ii) implies that $T_{W,\phi}(\nu)$ is well-defined and finite.

(i) This is pointed out in [6, Definition 1.6] by invoking [6, Lemma 2.2].

(ii), (iii) These are restatements of parts (i) and (ii) of [6, Theorem 1.6]. The fact that $\sup_{n\geq 1}Z_{n}(\theta)<\infty$ follows from the proof of [6, Corollary 1.3]. To prove $\Xi(F_{\theta})$ is compact, we invoke [6, Remark 2.1] to note that

\Xi(F_{\theta})=\arg\inf_{\nu\in\widetilde{\mathcal{M}}}J(\nu),

where the function $J(.)$ defined by

J(\nu):=D(\nu|\rho)-\theta T_{W,\phi}(\nu)

with $\phi(x_{1},\ldots,x_{v})=\prod_{a=1}^{v}x_{a}$ has compact level sets (by [6, Corollary 1.3, part (ii)]), and $\widetilde{\mathcal{M}}$ is a closed subset of probability measures.
∎

Remark 2.2.

We now claim that throughout the rest of the paper, without loss of generality we can assume that $W_{Q_{n}}$ converges to $W$ in strong cut metric $d_{\square}$ , instead of the weak cut metric $\delta_{\square}$ (see 1.1). Indeed, by definition of the weak-cut convergence, there exists a sequence of permutations $\{\pi_{n}\}_{n\geq 1}$ with $\pi_{n}\in S_{n}$ such that

d_{\square}(W_{Q_{n}^{\pi_{n}}},W)\rightarrow 0,\text{ where }Q_{n}^{\pi_{n}}(i,j):=Q_{n}(\pi_{n}(i),\pi_{n}(j)).

Then setting $Y_{i}=X_{\pi_{n}(i)}$ we have

	$\displaystyle\mathbb{U}_{n}({\bf x})=$	$\displaystyle\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in\mathcal{S}(n,v)}\left(\prod_{a=1}^{v}X_{i_{a}}\right)\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{a},i_{b})\right)$
	$\displaystyle=$	$\displaystyle\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in\mathcal{S}(n,v)}\left(\prod_{a=1}^{v}X_{\pi_{n}(i_{a})}\right)\left(\prod_{(a,b)\in E(H)}Q_{n}(\pi_{n}(i_{a}),\pi_{n}(i_{b}))\right)$
	$\displaystyle=$	$\displaystyle\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in\mathcal{S}(n,v)}\left(\prod_{a=1}^{v}Y_{i_{a}}\right)\left(\prod_{(a,b)\in E(H)}Q_{n}(\pi_{n}(i_{a}),\pi_{n}(i_{b}))\right)=:\widetilde{\mathbb{U}}_{n}({\bf x}).$

Set $\widetilde{\mathbb{R}}_{n,\theta}$ to be the Gibbs probability measure given by

\displaystyle\frac{d\widetilde{\mathbb{R}}_{n,\theta}}{d\mu^{\otimes n}}({\bf x})=\exp\Big{(}n\theta\widetilde{\mathbb{U}}_{n}({\bf x})-n\widetilde{Z}_{n}(\theta)\Big{)},

where $\widetilde{Z}_{n}(\theta)$ is the corresponding normalizing constant. Now if $(X_{1},\ldots,X_{n})\stackrel{{\scriptstyle IID}}{{\sim}}\mu$ , then so does $(X_{\pi_{n}(1)},\ldots,X_{\pi_{n}(n)})$ , and so

e^{n\widetilde{Z}_{n}(\theta)}=\mathbb{E}_{\mu^{\otimes n}}\exp\Big{(}n\theta\widetilde{U}_{n}({\bf X})\Big{)}=\mathbb{E}_{\mu^{\otimes n}}\exp\Big{(}n\theta{U}_{n}({\bf X})\Big{)}=e^{n{Z}_{n}(\theta)}.

Thus for any $\lambda\in\mathbb{R}$ we have

\mathbb{E}_{\widetilde{R}_{n,\theta}}\exp\Big{(}n\lambda\widetilde{U}_{n}({\bf X})\Big{)}=e^{n\widetilde{Z}_{n}(\theta+\lambda)-n\widetilde{Z}_{n}(\theta)}=e^{n{Z}_{n}(\theta+\lambda)-n{Z}_{n}(\theta)}=\mathbb{E}_{{R}_{n,\theta}}\exp\Big{(}n\lambda{U}_{n}({\bf X})\Big{)}.

In the above display, all quantities are finite and well defined using 1.1 part (ii). Thus the distribution of $\widetilde{\mathbb{U}}_{n}({\bf X})$ under $\widetilde{\mathbb{R}}_{n,\theta}$ is same as the distribution of ${\mathbb{U}}_{n}$ under ${\mathbb{R}}_{n,\theta}$ . Since $d_{\square}(W_{Q_{n}^{\pi_{n}}},W)\to 0$ , by replacing $W_{Q_{n}}$ by $W_{Q_{n}^{\pi_{n}}}$ without loss of generality we can assume $d_{\square}(W_{Q_{n}},W)\to 0$ as claimed, which we do throughout the rest of the paper.

Next, we state an elementary property of $\gamma(\cdot)$ that will be useful in proving 1.2 below. A short proof is provided in Section 5.

Lemma 2.2.

The function $\gamma\circ\beta(\cdot):cl(\mathcal{N})\to[0,\infty]$ is a continuous (possible extended) real-valued function.

Proof of 1.2.

(i) By switching the variables of integration, it is easy to check that the optimization problem (1.9) is equivalent to maximizing the function

\mathcal{G}_{W}(f):=\theta\int_{[0,1]^{v}}\mathrm{Sym}[W](x_{1},\ldots,x_{v})\left(\prod_{a=1}^{v}f(x_{a})\,dx_{a}\right)-\int_{[0,1]}\gamma(\beta(f(x)))dx.

Note that for all $\varepsilon\in[0,1]$ , $g\in\mathcal{L}$ and $f\in F_{\theta}\subseteq\mathcal{L}$ , the function $f+\varepsilon(f-g)=(1-\lambda)f+\varepsilon g\in\mathcal{L}$ , and so $\mathcal{G}_{W}(f+\varepsilon(g-f))\leq\mathcal{G}_{W}(f)$ . This gives

\frac{d}{d\varepsilon}\mathcal{G}_{W}(f+\varepsilon(g-f))\bigg{|}_{\varepsilon=0}\leq 0,

which is equivalent to

(2.1)

\displaystyle\footnotesize{\int_{[0,1]}\Big{(}(g(x_{1})-f(x_{1}))\underbrace{\left(\beta(f(x_{1}))-\theta v\int_{[0,1]^{v-1}}\mathrm{Sym}[W](x_{1},\ldots,x_{v})\left(\prod_{a=2}^{v}f(x_{a})\,dx_{a}\right)\right)}_{\delta(x_{1})}\,dx_{1}\Big{)}\geq 0.}`

We will show that $\lambda(\{x_{1}\in[0,1]:\ \delta(x_{1})\neq 0\})=0$ , where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$ . Let us assume the contrary. Without loss of generality, assume that $\lambda(\{x_{1}\in[0,1]:\ \delta(x_{1})>0\})>0$ . On this set, we have

f(x_{1})>\alpha^{\prime}\left(\theta v\int_{[0,1]^{v-1}}\mathrm{Sym}[W](x_{1},\ldots,x_{v})\left(\prod_{a=2}^{v}f(x_{a})\,dx_{a}\right)\right)=:v(x_{1}),

yielding

\lambda(\{x_{1}\in[0,1]:\ \delta(x_{1})>0,f(x_{1})>v(x_{1})\})>0.

This implies that there exists $\varepsilon>0$ such that

\lambda(\mathcal{A}_{\varepsilon})>0,\quad\mathcal{A}_{\varepsilon}:=\{x_{1}\in[0,1]:\ \delta(x_{1})>\epsilon,f(x_{1})>v(x_{1})+\varepsilon\})>0.

Define a function $g:[0,1]\mapsto{\rm cl}(\mathcal{N})$ by setting

g(x_{1}):=\begin{cases}f(x_{1})-\varepsilon&\mbox{if}\ x_{1}\in\mathcal{A}_{\epsilon},\\ f(x_{1})&\mbox{otherwise}.\end{cases}

Note that $g\in\mathcal{L}$ , as $f\in\mathcal{L}$ , and $\int(g(x_{1})-f(x_{1}))\delta(x_{1})dx_{1}<0$ , contradicting (2.1). This shows that $f(x_{1})=v(x_{1})$ a.s., as desired.

(ii) We will prove the contrapositive. Suppose there exists an almost surely constant function $f\in F_{\theta}^{(1)}$ , say $f(x)=c\neq 0$ for a.e. $x\in[0,1]$ . Then by (1.13), we have $c=\alpha^{\prime}(\theta c^{v-1}\mathcal{T}[\mathrm{Sym}[W]](x))$ for a.e. $x\in[0,1]$ . This implies $\mathcal{T}[\mathrm{Sym}[W]](\cdot)=\frac{\beta(c)}{\theta c^{v-1}}$ is constant almost surely, which is a contradiction.

(iii) Without loss of generality, assume that $\mathcal{T}[\mathrm{Sym}[W]](x)=1$ for a.e. $x\in[0,1]$ . Then $\mathrm{Sym}[W](x_{1},\ldots,x_{v})$ is a probability density function on $[0,1]^{v}$ with all marginals uniformly distributed on $[0,1]$ . By an application of Hölder’s inequality with respect to the probability measure induced by $\mathrm{Sym}[W]$ , we then have

\displaystyle\mathcal{G}_{W}(f)=\mathbb{E}_{(Z_{1},\ldots,Z_{v})\sim\mathrm{Sym}[W]}\bigg{[}\prod_{a=1}^{v}f(Z_{a})\bigg{]}\leq\int_{[0,1]}|f|^{v}(x)\,dx.

Consequently, it holds that

(2.2)

\displaystyle\sup_{f\in\mathcal{L}}\mathcal{G}_{W}(f)

\displaystyle\leq\sup_{f\in\mathcal{L}}\left\{\int_{0}^{1}[\theta|f|^{v}(x)-\gamma(\beta(f(x)))]\,dx\right\}\leq\sup_{t\in{\rm cl}(\mathcal{N})}\{\theta|t|^{v}-\gamma(\beta(t))\}.

(a) If $v$ is even, then (2.2) gives

\sup_{f\in\mathcal{L}}\mathcal{G}_{W}(f)\leq\sup_{t\in{\rm cl}(\mathcal{N})}\{\theta t^{v}-\gamma(\beta(t))\}.

Equality holds in the above display by taking $f$ to be constant functions. To find out the maximizing $f$ , we need equality in Hölder’s inequality. So $f$ must be a constant function a.s.
(b) If $\gamma(\beta(t))\leq\gamma(\beta(-t))$ for all $t\in\mathcal{N}\cap[0,\infty)$ , the same inequality continues to hold for all $t\in{\rm cl}(\mathcal{N})\cap(0,\infty)$ by 2.2. Thus (2.2) gives

\mathcal{G}_{W}(f)\leq\sup_{\begin{subarray}{c}t\in{\rm cl}(\mathcal{N}),t\geq 0\end{subarray}}\{\theta t^{v}-\gamma(\beta(t))\},

Again equality holds in the above display by taking supremum over constant functions, and the maximizing $f$ is again constant a.s.

(iv) The result for (a) follows immediately on noting that $\gamma(\beta(t))=\infty$ for all $t<0$ . We thus focus on proving (b). In this case there exists a symmetric measure $\nu$ such that $\mu=\nu_{B}$ . Fixing $t>0$ such that $-t\in\alpha^{\prime}(\mathbb{R})$ , using symmetry of $\nu$ it follows that $t\in\alpha^{\prime}(\mathbb{R})$ , and

\alpha(\theta)=\alpha_{\nu}(\theta+B)-\alpha_{\nu}(B),\text{ where }{\alpha}_{\nu}(\theta):=\log\int_{\mathbb{R}}e^{\theta x}d{\nu}(x)\text{ for all }\theta\in\mathbb{R}.

Thus, with ${\beta}_{\nu}$ denoting the inverse of ${\alpha}_{\nu}$ , we have ${\beta}_{\nu}(t)=\beta(t)+B$ for all $t\in\mathcal{N}_{\mu}$ , where ${\mathcal{N}}_{\nu}$ is the natural parameter space of ${\nu}$ . This gives

(2.3)

\displaystyle\gamma(\beta(t))=t{\beta}_{\nu}(t)-Bt-{\alpha}_{\nu}({\beta}_{\nu}(t))+{\alpha}_{\nu}(B)={\gamma}_{\nu}({\beta}_{\nu}(t))+{\alpha}_{\nu}(B)-Bt.

As ${\nu}$ is symmetric about $0$ , so ${\gamma}_{\nu}(\cdot)$ and ${\beta}_{\nu}(\cdot)$ are even functions. The assumption $B\geq 0$ along with (2.3) gives $\gamma(\beta(t))\leq\gamma(\beta(-t))$ for $t\geq 0$ , completing the proof. ∎

In the sequel, we will first prove 1.4 independently. Then we will prove 1.3 using 1.4. In order to prove 1.4, we need the following lemma whose proof we defer.

Lemma 2.3.

Suppose $\mu$ satisfies (1.6) for some $p>1$ .
(i) Then with $\alpha(\cdot)$ as in 1.3 we have

\lim_{\theta\to\pm\infty}\frac{\alpha^{\prime}(\theta)}{|\theta|^{\frac{1}{p-1}}}=0.

(ii) With $\beta(\cdot)$ as in 1.3 we have

\lim_{x\to\{\inf\{\mathcal{N}\},\sup\{\mathcal{N}\}\}}\frac{\beta(x)}{|x|^{p-1}}=\infty.

Proof of 1.4.

(i) (1.22) follows from [6, Eq 2.28].

We next prove (1.23). Fix $i\in[n]$ and use Hölder’s inequality to note that

	$\displaystyle\|m_{i}\|$	$\displaystyle\leq\frac{v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|\prod_{a=2}^{v}\|X_{i_{a}}\|$
		$\displaystyle\leq v\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{1}{q}}\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\prod_{a=2}^{v}\|X_{i_{a}}\|^{p}\right)^{\frac{1}{p}}$
(2.4)			$\displaystyle=v\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{1}{q}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p}}.$

Raising both sides to the $q^{th}$ power and summing (2.1) over $i\in[n]$ gives

	$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|^{q}$	$\displaystyle\leq v^{q}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{q(v-1)}{p}}\left(\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in[n]^{v}}\|\mathrm{Sym}[Q_{n}](i_{1},i_{2},\ldots,i_{v})\|^{q}\right)$
(2.5)			$\displaystyle\leq v^{q}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{q(v-1)}{p}}\\|W_{Q_{n}}\\|_{q\Delta}^{q\|E(H)\|},$

where the last inequality uses 2.1 part (c), with $W\equiv W_{Q_{n}}$ . The conclusion then follows by (1.8) and (1.22).

Next, we will prove (1.24). By 2.3 part (i), there exists $c_{\mu}>0$ such that for all $\theta\in\mathbb{R}$ we have

(2.6)

\displaystyle|\alpha^{\prime}(\theta)|\leq c_{\mu}|\theta|^{\frac{1}{p-1}}.

Now, note the following chain of equalities/inequalities with explanations to follow.

	$\displaystyle\|\alpha^{\prime}(\theta m_{i})\|$	$\displaystyle=\bigg{\|}\alpha^{\prime}\left(\frac{\theta v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in\mathcal{S}(n,v,i)}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|\prod_{a=2}^{v}\|X_{i_{a}}\|\right)\bigg{\|}$
		$\displaystyle\leq c_{\mu}\left(\frac{\theta v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|\prod_{a=2}^{v}\|X_{i_{a}}\|\right)^{\frac{1}{p-1}}$
		$\displaystyle\leq c_{\mu}(\theta v)^{\frac{1}{p-1}}\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{1}{q(p-1)}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p(p-1)}}.$

The first inequality follows directly from (2.6). The second inequality follows from (2.1). Raising both sides to the power $p$ and summing over $i\in[n]$ , we get:

	$\displaystyle\;\;\;\;\;\frac{1}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})\|^{p}$
	$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}c_{\mu}^{p}(\theta v)^{\frac{p}{p-1}}\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{p}{q(p-1)}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p-1}}$
(2.7)		$\displaystyle\leq c_{\mu}^{p}(\theta v)^{\frac{p}{p-1}}\left(1+\lVert W_{Q_{n}}\rVert_{q\Delta}^{q\|E(H)\|}\right)\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p-1}}.$

The final inequality follows by noting that $|x|^{\frac{p}{q(p-1)}}\leq 1+|x|$ and then using 2.1 part (c), with $W\equiv W_{Q_{n}}$ . The conclusion follows by (1.8) and (1.22).

(ii) The proof of (1.25) is very similar to the proof of (a). Firstly, $\sup_{\nu\in\Xi(F_{\theta})}\mathfrak{m}_{p}(\nu)<\infty$ follows from [6, Eq 2.29]. This also implies:

(2.8)

\sup_{f\in F_{\theta}}\lVert f\rVert_{p}=\sup_{\nu\in\Xi(F_{\theta})}\lVert\mathbb{E}_{\nu}[V|U]\rVert_{p}\leq\sup_{\nu\in\Xi(F_{\theta})}\mathbb{E}_{\nu}|V|^{p}=\sup_{\nu\in\Xi(F_{\theta})}\mathfrak{m}_{p}(\nu)<\infty.

Next, in the same vein as (2.1), we get

\sup_{\nu\in\mathfrak{B}_{\theta}^{*}}\mathfrak{m}_{q}(\nu)\leq v^{q}\sup_{f\in F_{\theta}}\lVert f\rVert_{p}^{q(v-1)}\lVert W\rVert_{q\Delta}^{q|E(H)|}<\infty,

by invoking (2.8) and (1.11), thereby proving the second conclusion. Finally, proceeding similar to (2.1), we have

\sup_{\nu\in\widetilde{\mathfrak{B}}_{\theta}}\mathfrak{m}_{p}(\nu)\leq c_{\mu}^{p}(\theta v)^{\frac{p}{p-1}}\left(1+\lVert W\rVert_{q\Delta}^{q|E(H)|}\right)\sup_{f\in F_{\theta}}\lVert f\rVert_{p}^{\frac{(v-1)p}{p-1}}<\infty,

where we have used (2.8) and (1.11). This completes the proof of part (b).

∎

2.2. Proof of 1.3

(i) The fact that $\Xi(F_{\theta})\subseteq\widetilde{\mathcal{M}}_{p}$ follows directly from (1.25). By an application of Hölder’s inequality with 2.1 part (iii), we get:

\mathbb{E}\left[\mathrm{Sym}[|W|](U_{1},U_{2},\ldots,U_{v})\prod_{a=2}^{v}|V_{a}|\right]\leq\lVert W\rVert_{q\Delta}^{|E(H)|}\left(\mathbb{E}|V_{1}|^{p}\right)^{\frac{v-1}{p}},

which is finite on using (1.6) and (1.11). By Fubini’s Theorem, $\vartheta_{W,\nu}(.)$ (see (1.18)) is well-defined a.s. on $[0,1]$ , as desired.

Remark 2.3.

Note that the above argument does not require $1/p+1/q<1$ but the weaker condition $1/p+1/q\leq 1$ .

(ii) We begin the proof with the following definition.

Definition 2.1.

Let $\mathcal{W}$ and $\widetilde{\mathcal{M}}_{p}$ be as in 1.1 and 1.2 respectively. Recall that $\mathfrak{m}_{p}(\nu)=\int|x|^{p}\,d\nu_{(2)}(x)<\infty$ (from 1.2) for $\nu\in\widetilde{\mathcal{M}}_{p}$ . Define

\mathcal{R}:=\{(W,\nu),\ W\in\mathcal{W},\ \nu\in\widetilde{\mathcal{M}}_{p},\ \lVert W\rVert_{q\Delta}<\infty\}.

Construct the following function $\Upsilon:\mathcal{R}\to\mathcal{M}$ (the space of probability measures on $[0,1]\times\mathbb{R}$ ) by setting:

(2.9)

\displaystyle\Upsilon(W,\nu):=\mathrm{Law}\left(U_{1},\vartheta_{W,\nu}(U_{1})\right).

Here $(U_{1},V_{1}),\ldots,(U_{v},V_{v})\overset{i.i.d.}{\sim}\nu$ , and $\vartheta_{W,\nu}(.)$ is as in (1.18). Note that $\Upsilon(W,\nu)$ is well-defined for $(W,\nu)\in\mathcal{R}$ , as the function $\vartheta_{W,\nu}(.)$ is well defined a.s. by 1.3 part (i). Also for $L>0$ and a random variable $X$ , set $X^{(L)}=X\mathbf{1}(|X|\leq L)$ . For any measure $\nu\in\widetilde{M}$ and $(U,V)\sim\nu$ , let $\nu^{(L)}$ denote the distribution of the truncated random variable $(U,V^{(L)})$ .

Set

(2.10)

m_{i,V}^{(L)}:=\frac{v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\mathrm{Sym}[Q_{n}]\left(i,i_{2},\ldots,i_{v}\right)\left(\prod_{a=2}^{v}X_{i_{a}}^{(L)}\right).

As a shorthand, we denote $m_{i,V}:=m_{i,V}^{\infty}$ . Let us also define

\mathbf{X}^{(L)}:=\{X_{1}^{(L)},\ldots,X_{n}^{(L)}\},\quad\mathbf{m}_{V}^{(L)}:=\{m_{1,V}^{(L)},\ldots,m_{n,V}^{(L)}\},\quad\mathbf{m}_{V}:=\{m_{1,V},\ldots,m_{n,V}\}.

Following (1.7), we have:

\mathfrak{L}_{n}(\mathbf{m}_{V}):=\frac{1}{n}\sum_{i=1}^{n}\delta_{\left(\frac{i}{n},m_{i,V}\right)}.

Next we generate $U\sim\mathrm{Unif}[0,1]$ . We define a map $\tilde{\mathfrak{L}}_{n}$ from $\mathbb{R}^{n}$ to $\widetilde{\mathcal{M}}$ given by

(2.11)

\tilde{\mathfrak{L}}_{n}(\mathbf{x}):=\mathrm{Law}(U,x_{\lceil nU\rceil}),\qquad\mathbf{x}=(x_{1},\ldots,x_{n}).

In view of (2.11), note that $\tilde{\mathfrak{L}}_{n}(\mathbf{X})$ , $\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})$ , $\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}^{(L)})$ , and $\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V})$ denotes the laws of $(U,X_{\lceil nU\rceil})$ , $(U,X_{\lceil nU\rceil}^{(L)})$ , $(U,m_{\lceil nU\rceil,V}^{(L)})$ , and $(U,m_{\lceil nU\rceil,V})$ conditioned on $X_{1},\ldots,X_{n}$ , respectively. Also, with $\Upsilon$ as in 2.1, we have

(2.12)

\Upsilon\left(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X})\right)=\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}),\quad\Upsilon\left(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})\right)=\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}^{(L)}).

In order to prove the above, note that, given any bounded continuous real-valued function $f$ on $[0,1]\times\mathbb{R}$ , we have:

	$\displaystyle\mathbb{E}_{\Upsilon\left(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X})\right)}[f]$
	$\displaystyle=\int_{0}^{1}f\left(u_{1},v\int_{[0,1]^{v-1}}\mathrm{Sym}[Q_{n}](\lceil nu_{1}\rceil,\lceil nu_{2}\rceil,\ldots,\lceil nu_{v}\rceil)\left(\prod_{a=2}^{v}X_{\lceil nu_{a}\rceil}\right)\,du_{2}\ldots\,du_{v}\right)\,du_{1}$
	$\displaystyle=\sum_{i_{1}=1}^{n}\int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}}f\left(u_{1},\frac{v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\mathrm{Sym}[Q_{n}](\lceil nu_{1}\rceil,i_{2},\ldots,i_{v})\left(\prod_{a=2}^{v}X_{i_{a}}\right)\right)\,du_{1}$
	$\displaystyle=\sum_{i_{1}=1}^{n}\int_{\frac{i_{1}-1}{n}}^{\frac{i_{1}}{n}}f(u_{1},m_{\lceil nu_{1}\rceil,V})\,du_{1}=\mathbb{E}_{\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V})}[f],$

and so the first conclusion of (2.12) holds. The proof of the second conclusion is similar. By the definition of $\mathfrak{B}^{*}_{\theta}$ in 1.3, we have

(2.13)

\mathfrak{B}^{*}_{\theta}=\{\Upsilon(W,\nu):\ \nu\in\Xi(F_{\theta})\}=\Upsilon(W,\Xi(F_{\theta})).

With $\mathfrak{L}_{n}(\mathbf{m})$ as in (1.15), triangle inequality gives

	$\displaystyle\;\;\;\;d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}),\mathfrak{B}^{*}_{\theta})$
	$\displaystyle\leq d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}),\mathfrak{L}_{n}(\mathbf{m}_{V}))+d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}_{V}),\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}))+d_{\ell}(\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}),\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}^{(L)}))$
	$\displaystyle\quad+d_{\ell}(\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}^{(L)}),\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})))+d_{\ell}(\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})),\mathfrak{B}^{*}_{\theta})$
	$\displaystyle=d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}),\mathfrak{L}_{n}(\mathbf{m}_{V}))+d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}_{V}),\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V}))+d_{\ell}(\Upsilon(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X})),\Upsilon(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})))$
	$\displaystyle\quad+d_{\ell}(\Upsilon(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})),\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)}))+d_{\ell}(\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})),\Upsilon(W,\Xi(F_{\theta})),$

where the second equality uses (2.12) and (2.13). We now show that each of the terms on the right hand side converge to $0$ as we take limits with $n\to\infty$ first, followed by $L\to\infty$ . Towards this direction, we observe that:

	$\displaystyle d_{\ell}\left(\mathfrak{L}_{n}(\mathbf{m}_{V}),\tilde{\mathfrak{L}}_{n}(\mathbf{m}_{V})\right)$	$\displaystyle=\sup_{f\in\mathrm{Lip}(1)}\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}f\left(\frac{i}{n},m_{i,V}\right)-\sum_{i=1}^{n}\int_{\frac{i-1}{n}}^{\frac{i}{n}}f(u,m_{i,V})\,du\bigg{\|}$
(2.14)			$\displaystyle\leq\sup_{f\in\mathrm{Lip}(1)}\sum_{i=1}^{n}\int_{\frac{i-1}{n}}^{\frac{i}{n}}\bigg{\|}f\left(\frac{i}{n},m_{i,V}\right)-f(u,m_{i,V})\bigg{\|}\,du\leq\frac{1}{n}\to 0.$

Based on the above two displays, it now suffices to prove the following:

(2.15)

d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}),\mathfrak{L}_{n}(\mathbf{m}_{V}))\overset{P}{\longrightarrow}0,

(2.16)

d_{\ell}(\Upsilon(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X})),\Upsilon(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})))\overset{P}{\longrightarrow}0,

as $n\to\infty$ followed by $L\to\infty$ , and

(2.17)

d_{\ell}(\Upsilon(W_{Q_{n}},\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})),\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})))\overset{P}{\longrightarrow}0,

as $n\to\infty$ for every fixed $L>0$ , and

(2.18)

d_{\ell}(\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})),\Upsilon(W,\Xi(F_{\theta}))\overset{P}{\longrightarrow}0,

as $n\to\infty$ , followed by $L\to\infty$ .

We now split the proof into four parts, proving the four preceding displays. We begin with the proof of (2.15) which requires the following lemma. It is a variant of [6, Lemma 2.7]. We omit the details of the proof for brevity.

Lemma 2.4.

Suppose $Q_{n}$ satisfies (1.8) for some $q>1$ . Let $\tilde{\phi}:\mathbb{R}^{v-1}\to[-L,L]$ for some $L>0$ , and $\mathcal{S}(n,v,i)$ be as in Definition 1.8. Then given any permutation $\sigma$ of $[v]$ , we get:

	$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{1}{n^{v}}\sup\limits_{\begin{subarray}{c}(x_{1},\ldots,x_{n})\\ \in\mathbb{R}^{n}\end{subarray}}\sum_{i_{1}=1}^{n}$	$\displaystyle\bigg{\lvert}\sum_{\begin{subarray}{c}(i_{2},\ldots,i_{v})\\ \in\mathcal{S}(n,v,i_{1})\end{subarray}}\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{\sigma(a)},i_{\sigma(b)})\right)\tilde{\phi}(x_{i_{2}},\ldots,x_{i_{v}})-$
		$\displaystyle\sum_{\begin{subarray}{c}(i_{2},\ldots,i_{v})\\ \in[n]^{v-1}\end{subarray}}\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{\sigma(a)},i_{\sigma(b)})\right)\tilde{\phi}(x_{i_{2}},\ldots,x_{i_{v}})\bigg{\rvert}=0.$

Proof of (2.15).

With $\mathcal{S}(n,v,i)$ as in Definition 1.8, define

(2.19)

m_{i}^{(L)}:=\frac{v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in\mathcal{S}(n,v,i)}\mathrm{Sym}[Q_{n}]\left(i,i_{2},\ldots,i_{v}\right)\left(\prod_{a=2}^{v}X_{i_{a}}^{(L)}\right).

It then suffices to prove the following:

(2.20)

\lim\limits_{L\to\infty}\limsup\limits_{n\to\infty}\mathbb{P}\left(n^{-1}\sum_{i=1}^{n}|m_{i}-m_{i}^{(L)}|\geq\epsilon\right)=0,

(2.21)

\lim\limits_{L\to\infty}\limsup\limits_{n\to\infty}\mathbb{P}\left(n^{-1}\sum_{i=1}^{n}|m_{i,V}-m_{i,V}^{(L)}|\geq\epsilon\right)=0,

for any $\epsilon>0$ , and for any fixed $L>0$ ,

(2.22)

\sum_{i\in[n]}|m_{i}^{(L)}-m_{i,V}^{(L)}|=o_{P}(n).

Proof of (2.20) Fix $\tilde{p}\in(1,p)$ such that $\tilde{p}^{-1}+q^{-1}<1$ . For any $L>1$ we have

	$\displaystyle\;\;\;\;\frac{1}{n}\sum_{i=1}^{n}\|m_{i}-m_{i}^{(L)}\|$
	$\displaystyle\leq\frac{v}{n^{v}}\sum_{A\subseteq\{2,\ldots,v\},\ \|A\|\geq 1}\sum_{(i_{1},\ldots,i_{v})\in[n]^{v}}\big{\|}\mathrm{Sym}[Q_{n}]\left(i_{1},i_{2},\ldots,i_{v}\right)\big{\|}\left(\prod_{a\in A}\|X_{i_{a}}-X_{i_{a}}^{(L)}\|\right)\left(\prod_{a\in A^{c}}\|X_{i_{a}}^{(L)}\|\right)$
	$\displaystyle\leq v\sum_{A\subseteq\{2,\ldots,v\},\ \|A\|\geq 1}\left[\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in[n]^{v}}\big{\|}\mathrm{Sym}[Q_{n}]\left(i_{1},i_{2},\ldots,i_{v}\right)\big{\|}^{q}\right]^{\frac{1}{q}}$
	$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left(\prod_{a\in A}\left(\frac{1}{n}\sum_{i_{a}=1}^{n}\|X_{i_{a}}\|^{\tilde{p}}\mathbf{1}(\|X_{i_{a}}\|>L)\right)\right)^{\frac{1}{\tilde{p}}}\left(\prod_{a\in A^{c}}\left(\frac{1}{n}\sum_{i_{a}=1}^{n}\|X_{i_{a}}\|^{\tilde{p}}\mathbf{1}(\|X_{i_{a}}\|\leq L)\right)\right)^{\frac{1}{\tilde{p}}}$
(2.23)		$\displaystyle\leq v2^{v-1}L^{\tilde{p}-p}\lVert W_{Q_{n}}\rVert_{q\Delta}^{\|E(H)\|}\left(1+\frac{1}{n}\sum_{i=1}^{n}\|X_{i}\|^{p}\right)^{\frac{v-1}{\tilde{p}}}.$

as $n\to\infty$ , followed by $L\to\infty$ . Here the second line uses (2.19), the third line uses Hölder’s inequality, and the fourth inequality follows from 2.1 part (iii) along with the inequalities

|x|^{\tilde{p}}\mathbf{1}(|x|>L)\leq L^{\tilde{p}-p}(1+|x|^{p}),\quad|x|^{\tilde{p}}\mathbf{1}(|x|\leq L)\leq 1+|x|^{p}.

The conclusion holds on noting that the RHS of (2.2) converges to $0$ on letting $n\to\infty$ followed by $L\to\infty$ , since $\lVert W_{Q_{n}}\rVert_{q\Delta}=O(1)$ and $\mathfrak{m}_{p}(\tilde{\mathfrak{L}}_{n}(\mathbf{X}))=O_{p}(1)$ (which are direct consequences of (1.8) and (1.22) respectively). This proves (2.20).

Proof of (2.21). The proof is same as that of (2.20). We skip the details for brevity.

Proof of (2.22). Using (2.22), observe that

	$\displaystyle\frac{1}{n}\sum_{i_{1}=1}^{n}\|m_{i_{1}}^{(L)}-m_{i_{1},V}^{(L)}\|$
	$\displaystyle=\frac{1}{n^{v}}\sum_{i_{1}=1}^{n}\Bigg{\|}\frac{1}{v!}\sum_{\sigma\in\mathcal{S}_{v}}\Bigg{[}\sum_{\begin{subarray}{c}(i_{2},\ldots,i_{v})\\ \in\mathcal{S}(n,v,i_{1})\end{subarray}}\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{\sigma(a)},i_{\sigma(b)})\right)\prod_{a=2}^{v}X_{i_{a}}^{(L)}$
	$\displaystyle\qquad-\sum_{\begin{subarray}{c}(i_{2},\ldots,i_{v})\\ \in[n]^{v-1}\end{subarray}}\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{\sigma(a)},i_{\sigma(b)})\right)\prod_{a=2}^{v}X_{i_{a}}^{(L)}\Bigg{]}\Bigg{\|}$
	$\displaystyle\leq\frac{1}{n^{v}}\max_{\sigma\in\mathcal{S}_{v}}\sum_{i_{1}=1}^{n}\bigg{\lvert}\sum_{\begin{subarray}{c}(i_{2},\ldots,i_{v})\\ \in\mathcal{S}(n,v,i_{1})\end{subarray}}\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{\sigma(a)},i_{\sigma(b)})\right)\prod_{a=2}^{v}X_{i_{a}}^{(L)}-$
	$\displaystyle\sum_{\begin{subarray}{c}(i_{2},\ldots,i_{v})\\ \in[n]^{v-1}\end{subarray}}\left(\prod_{(a,b)\in E(H)}Q_{n}(i_{\sigma(a)},i_{\sigma(b)})\right)\prod_{a=2}^{v}X_{i_{a}}^{(L)}\bigg{\rvert}.$

The RHS above converges to $0$ as $n\to\infty$ , using 2.4 with $\tilde{\phi}(x_{1},\ldots,x_{v})=\prod_{a=2}^{v}x^{(L)}_{a}$ along with triangle inequality.

∎

In order to prove (2.16) and (2.17), we need the following additional lemma whose proof we defer to Section 4.

Lemma 2.5.

Fix a graph $H$ with $v$ vertices and maximum degree $\Delta$ as before. Fix $p,q>0$ such that $\frac{1}{p}+\frac{1}{q}<1$ , $p\geq v$ . Then is well-defined on $\mathcal{R}$ , and the following conclusions hold:

(i)

Fix $C>0$ . Then

\lim\limits_{L\to\infty}\sup_{\nu\in\widetilde{\mathcal{M}}:\ \mathfrak{m}_{p}(\nu)\leq C}\sup_{W\in\mathcal{W}:\ \lVert W\rVert_{q\Delta}\leq C}d_{\ell}(\Upsilon(W,\nu),\Upsilon(W,\nu^{(L)}))=0.

(ii)

Suppose $W_{k},W_{\infty}\in\mathcal{W}$ , $k\geq 1$ such that $d_{\square}(W_{k},W)\to 0$ as $k\to\infty$ , and $\sup_{1\leq k\leq\infty}\lVert W_{k}\rVert_{q\Delta}<\infty$ for $1\leq k\leq\infty$ . Fix $L\in(0,\infty)$ and let $\widetilde{\mathcal{M}}^{(L)}$ denote the subset of $\widetilde{\mathcal{M}}$ for which the second marginal is compactly supported on $[-L,L]$ . Then we have

$\lim\limits_{k\to\infty}\sup_{\nu\in\widetilde{\mathcal{M}}^{(L)}}d_{\ell}(\Upsilon(W_{k},\nu),\Upsilon(W_{\infty},\nu))=0.$
(iii)

Fix $W\in\mathcal{W}$ such that $\lVert W\rVert_{q\Delta}<\infty$ , and let $\nu_{k},\nu_{\infty}\in\widetilde{\mathcal{M}}^{(L)}$ such that $d_{\ell}(\nu_{k},\nu_{\infty})\to 0$ . Then,

$\lim\limits_{k\to\infty}d_{\ell}(\Upsilon(W,\nu_{k}),\Upsilon(W,\nu_{\infty}))=0.$

Proof of (2.16).

We can use 2.5 part (i) to get the desired conclusion provided we can show $\lVert W_{Q_{n}}\rVert_{q\Delta}=O(1)$ and $\mathfrak{m}_{p}(\tilde{\mathfrak{L}}_{n}(\mathbf{X}))=O_{p}(1)$ (these requirements follow from the definition of $\mathcal{R}$ , see 2.1). But these are direct consequences of (1.8) and (1.22) respectively. ∎

Proof of (2.17).

We can use 2.5 part (ii) to get the desired conclusion, if we can verify that $d_{\square}(W_{Q_{n}},W)\to 0$ , $\lVert W_{Q_{n}}\rVert_{q\Delta}=O(1)$ , and $\lVert W\rVert_{q\Delta}=O(1)$ . But these are direct consequences of (1.4), (1.8), and (1.11) respectively. ∎

The final step is to establish (2.18) for which we need two results. The first one is an immediate corollary of 2.5 parts (i) and (iii) (and hence it’s proof is omitted), while the second one is a simple convergence lemma, whose proof is provided in Section 5.2.

Corollary 2.6.

Consider the same setting as in 2.5. For $C>0$ , define

(2.24)

\displaystyle\widetilde{\mathcal{M}}_{p,C}:=\{\nu\in\widetilde{\mathcal{M}}_{p}:\ \mathfrak{m}_{p}(\nu)\leq C\}.

Suppose $W\in\mathcal{W}$ be such that $\lVert W\rVert_{q\Delta}<\infty$ , then $\Upsilon(W,\cdot)$ is continuous on $\widetilde{\mathcal{M}}_{p,C}$ in the weak topology.

Lemma 2.7.

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ be two Polish spaces. Let $\xi_{n}$ be a sequence of $X$ -valued random variables such that $d_{X}(\xi_{n},\mathcal{F})\xrightarrow{\text{P}}0$ for some closed set $\mathcal{F}\subseteq X$ . Assume that there exists a compact set $K\subseteq X$ such that

(2.25)

\displaystyle\lim\limits_{n\to\infty}\mathbb{P}(\xi_{n}\notin K)=0.

Finally consider a function $g:X\mapsto Y$ such that $g$ is continuous on $K$ . Then we have

d_{Y}(g(\xi_{n}),g(\mathcal{F}))\xrightarrow{\text{P}}0.

Proof of (2.18).

Applying 2.5 part (i), for every $\varepsilon>0$ we have

\lim_{L\to\infty}\limsup_{n\to\infty}\;\;\mathbb{P}\left(d_{\ell}(\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X}^{(L)})),\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X})))\geq\varepsilon\right)=0.

It thus suffices to show that

(2.26)

d_{\ell}(\Upsilon(W,\tilde{\mathfrak{L}}_{n}(\mathbf{X})),\Upsilon(W,\Xi(F_{\theta})))\overset{P}{\longrightarrow}0.

To this effect, use 1.1 part (iv) to note that

(2.27)

\displaystyle d_{\ell}(\tilde{\mathfrak{L}}_{n}(\mathbf{X}),\Xi(F_{\theta}))\stackrel{{\scriptstyle P}}{{\to}}0,

where the set $\Xi(F_{\theta})$ is compact in the weak topology. Also note that by (1.22), there exists $C>0$ such that

(2.28)

\displaystyle\lim\limits_{n\to\infty}\mathbb{P}(\tilde{\mathfrak{L}}_{n}(\mathbf{X})\notin\widetilde{\mathcal{M}}_{p,C})=0.

We will now invoke 2.7 with $X=\widetilde{\mathcal{M}}_{p}$ and $Y=\mathcal{M}$ , both coupled with weak topology, $\xi_{n}=\tilde{\mathfrak{L}}_{n}(\mathbf{X})$ , $\mathcal{F}=\Xi(F_{\theta})$ , $K=\widetilde{\mathcal{M}}_{p,C}$ and $g(\cdot)=\Upsilon(W,\cdot)$ . Once we verify the conditions of 2.7 with the above specifications, we will then conclude (2.26), which in turn, completes the proof.

To verify the conditions of 2.7, note that $\mathcal{F}=\Xi(F_{\theta})$ is compact, and is a subset of $X=\widetilde{\mathcal{M}}_{p}$ by (1.25). Further, (2.27) implies $d_{X}(\xi_{n},\mathcal{F})\overset{P}{\to}0$ . The conclusion in (2.28) implies (2.25). The fact that $g(\cdot)=\Upsilon(W,\cdot)$ is well-defined on $X$ follows from 1.3 part (a) (also see 2.1). Finally, the continuity of $g$ on $K$ follows from 2.6.

This finally completes the proof of 1.3. ∎

2.3. Proofs of 1.5 and 1.6

In order to prove 1.5, we need the following results. The first result is a lemma about a sequence of functions converging in measure. It’s proof is deferred to Section 5.2.

Lemma 2.8.

Let $U\sim\mathrm{Unif}[0,1]$ and $p\geq 1$ .

(i)

Suppose $\{f_{n}\}_{n\geq 1}$ is a sequence of measurable real-valued functions on $[0,1]$ such that

\limsup_{n\to\infty}\mathbb{E}|f_{n}(U)|^{p}<\infty,\text{ and }(U,f_{n}(U))\overset{D}{\longrightarrow}(U,f_{\infty}(U)).

Then for any $\tilde{p}\in(0,p)$ we have:

(2.29)

\mathbb{E}|f_{n}(U)-f_{\infty}(U)|^{\tilde{p}}\longrightarrow 0.

(ii)

If $(U,f(U))\overset{D}{=}(U,g(U))$ for some $f,g$ such that $\mathbb{E}|f(U)|^{p}<\infty$ and $\mathbb{E}|g(U)|^{p}<\infty$ , then $f(U)=g(U)$ a.s.

For stating the second result, we recall the definitions of $\mathfrak{B}^{*}_{\theta}$ , $\widetilde{\mathfrak{B}}_{\theta}$ , $\vartheta_{W,\nu}$ , $\Xi(F_{\theta})$ , $\mathcal{L}$ from 1.3, (1.21), (1.18), 1.1 part (iv), and 1.4, respectively. Also define

(2.30)

\widetilde{M}_{p,C}:=\left\{\mathrm{Law}(U,f(U)):\ f\in\mathcal{L},\ \int_{0}^{1}|f(u)|^{p}\,du\leq C\right\},\;\;\widetilde{M}_{p}:=\cup_{C\in\mathbb{N}}\ \widetilde{M}_{p,C}.

Based on 1.2 and (2.24), $\widetilde{M}_{p}\subseteq\widetilde{\mathcal{M}}_{p}$ and $\widetilde{M}_{p,C}\subseteq\widetilde{\mathcal{M}}_{p,C}$ . We also construct $G_{1}:[0,1]\times\mathbb{R}\to[0,1]\times\mathcal{N}$ given by $G_{1}(x,y):=(x,\alpha^{\prime}(\theta y))$ .

We note an elementary observation here which will be used in the sequel. To wit, recall from 1.3 that $\mathfrak{B}_{\theta}^{*}=\{\mathrm{Law}(U,\vartheta_{W,\nu}(U)),\nu\in\Xi(F_{\theta})\}$ . Consequently from the definition of $G_{1}$ it follows that:

(2.31)

\displaystyle\widetilde{\mathfrak{B}}_{\theta}=G_{1}(\mathfrak{B}_{\theta}^{*}):=\{{\rm Law}(U,\alpha^{\prime}(\theta\vartheta_{W,\nu}(U))),\nu\in\Xi(F_{\theta})\}.

We now state the following lemma, formalizes a key property of the sets $\mathfrak{B}_{\theta}^{*}$ and $\widetilde{\mathfrak{B}}_{\theta}$ . It’s proof is deferred to Section 4.

Lemma 2.9.

Consider the same setting as in 1.3. Then the set $\mathfrak{B}^{*}_{\theta}$ is a compact subset of $\widetilde{M}_{q}$ in the weak topology, whereas $\widetilde{\mathfrak{B}}_{\theta}$ is a compact subset of $\widetilde{M}_{p}$ in the weak topology.

We are now in the position to prove 1.5.

Proof of 1.5.

By arguments similar to (2.2) we have

d_{\ell}(\mathfrak{L}_{n}(\mathbf{m}),\tilde{\mathfrak{L}}_{n}(\mathbf{m}))\overset{P}{\longrightarrow}0.

Consequently by invoking 1.3 part (b) we get:

(2.32)

\displaystyle d_{\ell}(\tilde{\mathfrak{L}}_{n}(\mathbf{m}),\mathfrak{B}_{\theta}^{*})\overset{P}{\longrightarrow}0.

Further by (1.23), there exists $C>0$ such that

(2.33)

\displaystyle\lim\limits_{n\to\infty}\mathbb{P}(\tilde{\mathfrak{L}}_{n}(\mathbf{m})\notin\widetilde{M}_{q,C})=0.

With the above observations in mind, we invoke 2.7 with $X=\widetilde{M}_{q}$ , $Y=\mathcal{M}$ equipped with the topology of weak convergence, $\xi_{n}=\tilde{\mathfrak{L}}_{n}(\mathbf{m})$ , $\mathcal{F}=\mathfrak{B}_{\theta}^{*}$ , $g=G_{1}$ and $K=\widetilde{M}_{q,C}$ (with $C$ chosen as in (2.33)). Once we verify the assumptions of 2.7 with the above specifications, by (2.31), we obtain:

(2.34)

d_{\ell}\left(\tilde{\mathfrak{L}}_{n}(\boldsymbol{\alpha}),\tilde{\mathfrak{B}}_{\theta}\right)\overset{P}{\longrightarrow}0,

To verify the conditions of 2.7, note that $\mathcal{F}=\mathfrak{B}_{\theta}^{*}\subseteq X=\widetilde{M}_{q}$ by (1.25). Further, (2.32) implies $d_{X}(\xi_{n},\mathcal{F})\overset{P}{\to}0$ and 1.4 part (ii) along with Fatou’s lemma implies $\mathcal{F}$ is a compact subset of $X$ . The conclusion in (2.33) implies (2.25). Finally, the continuity of $g$ on $K$ follows from the continuity of $\alpha^{\prime}(\cdot)$ .

We now use (2.34) to complete the proof. The key tool will once again be 2.7. To set things up, fixing $C>0$ equip $\widetilde{M}_{p,C}$ with the weak topology. Pick any $\nu\in\widetilde{M}_{p,C}$ . Then $\nu$ is distributed as $(U,f(U))$ , where $U\sim\mathrm{Unif}[0,1]$ and $f:[0,1]\mapsto\mathbb{R}$ is measurable with $\lVert f\rVert_{p}\leq C$ . Consequently by 2.8 part (ii), the map $G_{2}:\widetilde{\mathcal{M}}_{p,C}\to L^{p^{\prime}}[0,1]$ , (for some $p^{\prime}<p$ ) given by $G_{2}(\nu)=f$ is well-defined.

For any $f\in F_{\theta}$ , setting $\nu=\Xi(f)$ use (1.19) to note that $f(U)=\alpha^{\prime}(\theta\vartheta_{W,\nu}(U))$ a.s. Consequently, by (2.31), we get:

(2.35)

\displaystyle G_{2}(\widetilde{\mathfrak{B}}_{\theta})=F_{\theta}.

Moreover, by (1.24), there exists $C>0$ such that

(2.36)

\displaystyle\lim\limits_{n\to\infty}\mathbb{P}(\tilde{\mathfrak{L}}_{n}(\boldsymbol{\alpha})\notin\widetilde{M}_{p,C})=0.

With this observation, we will invoke 2.7 with $X=\widetilde{M}_{p}$ , $Y\equiv L^{p^{\prime}}[0,1]$ , equipped with the topologies of weak convergence and $L^{p^{\prime}}[0,1]$ respectively, and $\xi_{n}=\tilde{\mathfrak{L}}_{n}(\boldsymbol{\alpha})$ , $\mathcal{F}\equiv\widetilde{\mathfrak{B}}_{\theta}$ , $g\equiv G_{2}$ , and $K=\widetilde{M}_{p,C}$ with $C$ chosen from (2.36). Once we verify the conditions of 2.7, an application of (2.35) will yield

\lVert G_{2}(U,\alpha^{\prime}(\theta m_{\lceil nU\rceil}))-G_{2}(\widetilde{\mathfrak{B}}_{\theta})\rVert_{p^{\prime}}=\inf_{f\in F_{\theta}}\int_{0}^{1}|\alpha^{\prime}(\theta m_{\lceil nu\rceil})-f(u)|^{p^{\prime}}\,du\overset{P}{\longrightarrow}0,

which will complete the proof of (1.26).

To verify the conditions of 2.7, note that $\mathcal{F}=\widetilde{\mathfrak{B}}_{\theta}\subseteq X=\widetilde{M}_{p}$ by (1.25). Further, (2.34) implies $d_{X}(\xi_{n},\mathcal{F})\overset{P}{\to}0$ , and 1.4 part (ii) implies $\mathcal{F}$ is a compact subset of $X$ . The conclusion in (2.36) implies (2.25). The fact that $g(\cdot)=G_{2}(\cdot)$ is well-defined on $X=\widetilde{M}_{p}$ follows from 2.8 part (b). Continuity of $g$ on $K$ follows from 2.8 part (i).

∎

For proving 1.6, we will need the following lemma whose proof we defer to Section 4.

Lemma 2.10.

Suppose ${\bf X}$ is a sample from the model (1.3) ( $\theta$ need not be non-negative). Suppose $p\in[v,\infty]$ , $q>1$ satisfy (1.6), $\limsup_{n\to\infty}\lVert W_{Q_{n}}\rVert_{q\Delta}<\infty$ and $\frac{1}{p}+\frac{1}{q}\leq 1$ .

(i) Given any vector $\mathbf{d}^{(N)}:=(d_{1},d_{2},\ldots,d_{N})$ such that $\lVert\mathbf{d}^{(N)}\rVert_{\infty}=O(1)$ , we have

\sum_{i=1}^{N}d_{i}(X_{i}-\alpha^{\prime}(\theta m_{i}))=o_{p}(n).

(ii) If $\frac{1}{p}+\frac{1}{q}<1$ , then

\sum_{i=1}^{n}m_{i}\left(X_{i}-\alpha^{\prime}(\theta m_{i})\right)=o_{P}(n).

Proof of 1.6.

By 1.2 part (iii), all the optimizers of the problem in (1.9) are constant functions. Further, (1.25) shows that there exists $K>0$ (depending on $\theta$ ) such that all the optimizers of (1.9) have $L^{p}$ norm bounded by $K$ . Combining these two observations, we have that $F_{\theta}$ consists only of constant functions where the constants are given by

\mathcal{A}_{\theta}=\operatorname{argmin}_{t\in\mathcal{N},\ |t|\leq K}[\gamma(\beta(t))-\theta t^{v}].

As analytic non-constant functions can only have finitely many optimizers in a compact set, it follows that $A_{\theta}$ is a finite set.

(i) Define $c_{i,L}:=c_{i}1\{|c_{i}|\leq L\}$ and $\bar{m}:=n^{-1}\sum_{i=1}^{n}m_{i}$ . We claim that result follows given the following display:

(2.37)

\lim_{L\to\infty}\limsup_{n\to\infty}\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\left|c_{i}-c_{i}^{(L)}\right||X_{i}|\right]=0.

This is because given any $L>0$ and any $t\in\mathcal{A}_{\theta}$ (recall this implies $|t|\leq K$ ), the following inequalities hold:

	$\displaystyle\;\;\;\;\;\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}c_{i}X_{i}\bigg{\|}$
	$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\bigg{\|}c_{i}-c_{i}^{(L)}\bigg{\|}\|X_{i}\|+\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}c_{i}^{(L)}(X_{i}-\alpha^{\prime}(\theta m_{i}))\bigg{\|}+\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}c_{i}^{(L)}(\alpha^{\prime}(\theta m_{i})-t)\bigg{\|}+\frac{\|t\|}{n}\bigg{\|}\sum_{i=1}^{n}c_{i}^{(L)}\bigg{\|}$
	$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\bigg{\|}c_{i}-c_{i}^{(L)}\bigg{\|}\|X_{i}\|+\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}c_{i}^{(L)}(X_{i}-\alpha^{\prime}(\theta m_{i}))\bigg{\|}+\frac{L}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})-t\|$
	$\displaystyle\quad\quad\quad+\frac{K}{n}\bigg{\|}\sum_{i=1}^{n}c_{i}\bigg{\|}+\frac{K}{nL^{r-1}}\sum_{i=1}^{n}\|c_{i}\|^{r}.$

Taking an infimum over $t\in\mathcal{A}_{\theta}$ gives the bound

	$\displaystyle\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}c_{i}X_{i}\bigg{\|}$	$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\bigg{\|}c_{i}-c_{i}^{(L)}\bigg{\|}\|X_{i}\|+\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}c_{i}^{(L)}(X_{i}-\alpha^{\prime}(\theta m_{i}))\bigg{\|}$
		$\displaystyle+\inf_{t\in\mathcal{A}_{\theta}}\frac{L}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})-t\|+\frac{K}{n}\bigg{\|}\sum_{i=1}^{n}c_{i}\bigg{\|}+\frac{K}{nL^{r-1}}\sum_{i=1}^{n}\|c_{i}\|^{r}.$

The first and last terms above converge to $0$ in probability as $n\to\infty$ first, followed by $L\to\infty$ , by using (2.37) and the assumption $\sum_{i=1}^{n}|c_{i}|^{r}=O(n)$ for $r>1$ . The remaining terms converge to $0$ as $n\to\infty$ for fixed $L>0$ , by using 2.10 part (i), (1.26), and $\sum_{i=1}^{n}c_{i}=o(n)$ , respectively. This completes the proof.

Next, we prove (2.37). Fix $\tilde{r}\in(1,r)$ such that $\frac{1}{p}+\frac{1}{\tilde{r}}=1$ . By Hölder’s inequality,

		$\displaystyle\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\left\|c_{i}-c_{i}^{(L)}\right\|\|X_{i}\|\right]=\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\left\|c_{i}\right\|1\{\|c_{i}\|>L\}\|X_{i}\|\right]$
	$\displaystyle\leq$	$\displaystyle\left(\frac{1}{N}\sum_{i=1}^{n}\|c_{i}\|^{\tilde{r}}1\{\|c_{i}\|>L\}\right)^{\frac{1}{\tilde{r}}}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\|X_{i}\|^{p}\right)^{\frac{1}{p}}\leq\frac{1}{L^{\frac{r-\tilde{r}}{\tilde{r}}}}\left(\frac{1}{N}\sum_{i=1}^{n}\|c_{i}\|^{r}\right)^{\frac{1}{\tilde{r}}}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\|X_{i}\|^{p}\right)^{\frac{1}{p}}.$

This, along with (1.22) and the assumption $\sum_{i=1}^{n}|c_{i}|^{r}=O(n)$ , establishes (2.37).

(ii) As in part (i), we have

\displaystyle\inf_{t\in\mathcal{A}_{\theta}}\bigg{|}\frac{1}{n}\sum_{i=1}^{n}c_{i}X_{i}-\tilde{c}t\bigg{|}\leq\bigg{|}\frac{1}{n}\sum_{i=1}^{n}(c_{i}-\tilde{c})X_{i}\bigg{|}+\frac{|\tilde{c}|}{n}\bigg{|}\sum_{i=1}^{n}(X_{i}-\alpha^{\prime}(\theta m_{i}))\bigg{|}+\frac{|\tilde{c}|}{n}\inf_{t\in\mathcal{A}_{\theta}}\sum_{i=1}^{n}|\alpha^{\prime}(\theta m_{i})-t|.

The first term converges to $0$ in probability by part (i), the second converges to $0$ by 2.10 part (a), and the third term converges to $0$ by (1.26). This completes the proof.

(iii) We begin by observing that for any $L>1$ , we have:

\mathfrak{C}_{L}:=\sup_{x\in[-L,L]}\bigg{|}\frac{d}{dx}(x\alpha^{\prime}(x))\bigg{|}<\infty,\quad\quad\widetilde{\mathfrak{C}}_{L}:=\inf_{x\in[-L,L]}\alpha^{\prime\prime}(x)>0,

both of which follow from standard properties of exponential families. Recall that for any $t\in\mathcal{A}_{\theta}$ , we have $t=\alpha^{\prime}(\theta vt^{v-1})$ by 1.2 part (i). This gives

|\alpha^{\prime}(\theta m_{i})-t|=|\alpha^{\prime}(\theta m_{i})-\alpha^{\prime}(\theta vt^{v-1})|\geq|\theta|\widetilde{\mathfrak{C}}_{L|\theta|}|m_{i}-vt^{v-1}|,

and so for all large $L$ (depending on $\theta,K$ ) we have

	$\displaystyle\;\;\;\;\inf_{t\in\mathcal{A}_{\theta}}\frac{1}{n}\sum_{i=1}^{n}\|m_{i}-vt^{v-1}\|$
	$\displaystyle\leq\inf_{t\in\mathcal{A}_{\theta}}\frac{1}{n\widetilde{\mathfrak{C}}}_{{L\|\theta\|}}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})-t\|+\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|\mathbbm{1}(\|m_{i}\|\geq L)$
(2.38)		$\displaystyle\leq\inf_{t\in\mathcal{A}_{\theta}}\frac{1}{n\widetilde{\mathfrak{C}}_{L}\|\theta\|}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})-t\|+\frac{1}{nL^{q-1}}\sum_{i=1}^{n}\|m_{i}\|^{q}.$

The RHS above converges to $0$ in probability, as $n\to\infty$ , followed by $L\to\infty$ . This is because, the first term in (2.3) converges to $0$ in probability as $n\to\infty$ for every fixed $L$ , by using (1.26). The second term converges to to $0$ in probability by taking $n\to\infty$ first, followed by $L\to\infty$ , by using (1.23).

Next choose $\tilde{q}\in(1,q)$ such that $p^{-1}+\tilde{q}^{-1}=1$ . Note that for any $t\in\mathcal{A}_{\theta}$ , $vt^{v-1}\alpha^{\prime}(\theta vt^{v-1})=vt^{v}$ (using the relation $t=\alpha^{\prime}(\theta vt^{v-1})$ ). In the same vein as (2.3), by using (1.17), we also get for all $L$ large enough:

		$\displaystyle\;\;\;\;\;\inf_{t\in\mathcal{A}_{\theta}}\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\alpha^{\prime}(\theta m_{i})-vt^{v}\|$
(2.39)			$\displaystyle\leq\inf_{t\in\mathcal{A}_{\theta}}\frac{\mathfrak{C}_{L\|\theta\|}}{\|\theta\|n}\sum_{i=1}^{n}\|m_{i}-vt^{v-1}\|+\frac{1}{L^{\frac{q}{\tilde{q}}-1}}\left(\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|^{q}\right)^{\frac{1}{\tilde{q}}}\left(\frac{1}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})\|^{p}\right)^{\frac{1}{p}}.$

The RHS above converges to $0$ in probability, as $n\to\infty$ , followed by $L\to\infty$ . This is because, the first term converges to $0$ as $n\to\infty$ by (2.3), and the second term converges to $0$ as $n\to\infty$ followed by $L\to\infty$ by using (1.23) and (1.22). Finally, the conclusion in part (iii) follows by combining (2.3) with 2.10 part (ii).

∎

3. Proof of Results from Section 1.2

Proof of 1.8.

(i) Note that quadratic forms correspond to the choice $H=K_{2}$ and $v=2$ in (1.2). Let $\mu_{\theta}$ be the tilted probability measure on $\mathbb{R}$ obtained from $\mu$ as in 1.3. Then a direct computation using (1.28) gives

(3.1)

\displaystyle Z^{\mathrm{quad}}_{n}(\theta,B)={\alpha}(B)+\frac{1}{n}\log{\mathbb{E}_{{\bf X}\sim{\mu}_{B}^{\otimes n}}\exp\left(\frac{\theta}{n}\sum_{i\neq j}Q_{n}(i,j)X_{i}X_{j}\right)}.

Using this along with 1.1 part (iii) we get

	$\displaystyle Z^{\mathrm{quad}}_{n}(\theta,B)-\alpha(B)$
	$\displaystyle\to\sup_{f\in\mathcal{L}}\left(\theta\int_{[0,1]^{2}}W(x,y)f(x)f(y)\,dx\,dy-\int_{[0,1]}{\gamma}_{B}({\beta}_{B}(f(x)))\,dx\right)$
(3.2)		$\displaystyle=\sup_{f\in\mathcal{L}}\left(\theta\int_{[0,1]^{2}}W(x,y)f(x)f(y)\,dx\,dy-\int_{[0,1]}(\gamma(\beta(f(x)))+{\alpha}(B)-Bf(x))\,dx\right).$

Here ${\gamma}_{B}(.)$ and ${\alpha}_{B}(.)$ are as in 1.3, but for the tilted measure ${\mu}_{B}$ instead of $\mu$ , and the last equality uses (2.3). By invoking 1.2 part (iii), if $v$ is even, the set of optimizers $F_{\theta}\equiv F_{\theta,B}$ in the above display are constant functions, where the constant is an optimizer of the following optimization problem:

(3.3)

\sup_{x\in\alpha^{\prime}(\mathbb{R})}\left(\theta x^{2}+Bx-x\beta(x)+\alpha(\beta(x))\right).

By 1.7 (parts (i) and (ii)), if either (a) $B\neq 0$ , or (b) $B=0$ , $\theta\leq(\alpha^{\prime\prime}(0))^{-1}/2$ , then the optimizer is $x=t_{\theta,B,\mu}$ . On the other hand when $B=0$ and $\theta>(\alpha^{\prime\prime}(0))^{-1}/2$ , by 1.7 part (iii) the optimizers are $x=\pm t_{\theta,B,\mu}$ . Using this, the desired conclusion of part (i) follows.

(ii) Recall the definition of $\mathcal{A}_{\theta}\equiv\mathcal{A}_{\theta,B}$ from (1.27), and use part (i) to note that all functions in $F_{\theta,B}$ are constant functions, with constants belonging to the set $\mathcal{A}_{\theta,B}$ . Since $v=2$ , we have

\{{vt^{v}}:\ t\in\mathcal{A}_{\theta,B}\}=\{{2t^{2}}:\ t\in\mathcal{A}_{\theta,B}\}=2t_{\theta,B}^{2}\Rightarrow\frac{1}{n}\sum_{i=1}^{n}X_{i}m_{i}\overset{d}{\longrightarrow}2t_{\theta,B,\mu}^{2},

where we use 1.6 part (iii).

For the weak limit of $\bar{X}$ we invoke 1.6 part (ii) with $c_{i}=1$ which implies $\tilde{c}=1$ . The conclusion follows by noting that when $B=0$ , the symmetry of $\mu$ about the origin implies that $\bar{X}$ and $-\bar{X}$ have the same distribution. ∎

Proof of 1.9.

(i) Let $\mu_{\theta}$ be the tilted measure obtained from $\mu$ as in 1.3, and let $\alpha_{B}(.),\beta_{B}(.),{\gamma}_{B}(.)$ be as in 1.3, but for the measure ${\mu}_{B}$ instead of $\mu$ . Using (2.3) we get

\gamma_{B}(\beta_{B}(t))=\gamma(\beta(t))+\alpha(B)-Bt,

using which the optimization problem in (1.32) (ignoring the additive constant $\alpha(B)$ ) becomes

(3.4)

\displaystyle\sup_{f\in\mathcal{L}:\ \int_{[0,1]}\gamma(\beta(f(x)))dx<\infty}\left\{\theta G_{W}(f)-\int_{[0,1]}\gamma_{B}(\beta_{B}(f(x)))dx\right\}.

Now, we invoke 1.2 part (i) to conclude that any maximizer of the above display satisfies the fixed point equation (1.33).
(ii) It suffices to show that all optimizers of (3.4) are constant functions, for which invoking 1.2 part (iii) it suffices to show that $\mu_{B}$ is stochastically non-negative (as per 1.7), if $\mu$ is stochastically non-negative. In this case we have $\gamma(\beta(t))\leq\gamma(\beta(-t))$ for $t\geq 0$ . Along with (2.3), this gives

\gamma_{B}(\beta_{B}(t))=\gamma(\beta(t))+\alpha(B)-Bt\leq\gamma(\beta(-t))+\alpha(B)+Bt=\gamma_{B}(\beta_{B}(-t)),

where we use the fact that $B\geq 0$ . This shows that $\mu_{B}$ is stochastically non-negative as well.

Hence, by the proof of 1.2 part (iii), the maximizers of (1.32) are constant functions provided either $v$ is even or $\mu$ is stochastically non-negative. Finally, (1.33) follows from (1.32), on setting $f(.)$ to be a constant function.

(iii)(a) The optimization problem (1.32) reduces to maximizing

(3.5)

\displaystyle H_{\theta,B}(x):=\theta x^{v}+Bx-\gamma(\beta(x))

over $x\in[-1,1]$ . Differentiating we get

(3.6)

\displaystyle H^{\prime}_{\theta,B}(x)=\theta vx^{v-1}+B-\beta(x),\quad H^{\prime\prime}_{\theta,B}(x)=\theta v(v-1)x^{v-2}-\beta^{\prime}(x).

Since, $\mu$ is supported on $[-1,1]$ , we have $\lim_{\theta\to\infty}\alpha^{\prime}(\theta)=1$ , and so

\alpha^{\prime\prime}(\theta)=\mathbb{E}_{\mu_{\theta}}(X^{2})-(\alpha^{\prime}(\theta))^{2}\leq 1-(\alpha^{\prime}(\theta))^{2}\rightarrow 0

as $\theta\rightarrow\infty$ . Hence, there exists $B_{0}=B_{0}(\theta,v)$ such that for $B\geq B_{0}$ we have $\alpha^{\prime\prime}(B)<\frac{1}{2\theta v(v-1)}$ . If $x$ is a global maximizer of $H(.)$ , then we have

x=\alpha^{\prime}(\theta vx^{v-1}+B)\geq\alpha^{\prime}(B)\implies\beta(x)\geq B.

However, on the interval $\{x:\beta(x)\geq B\}$ , using boundedness of support, we have

H^{\prime\prime}_{\theta,0}(x)\leq\theta v(v-1)-\frac{1}{\alpha^{\prime\prime}(\beta(x))}<0.

Thus $H_{\theta,B}(.)$ is strictly concave on the interval $\{x:\beta(x)\geq B\}$ , and so the global maximizer must be unique.

(iii)(b) We break the proof into the following steps:

•

There exists $\theta_{1c}\in(0,\infty)$ such that $0$ is the unique global maximizer for $H_{\theta,0}(.)$ .
Since $\mu$ is compactly supported on $[-1,1]$ , we have

\alpha^{\prime\prime}(\theta)=\mathrm{Var}_{\mu_{\theta}}(X)\leq 1,\text{ and so }\beta^{\prime}(x)=\frac{1}{\alpha^{\prime\prime}(\beta(x))}\geq 1.

Thus for $\theta<\frac{1}{2v(v-1)}=:\theta_{1c}$ we have

\displaystyle H^{\prime\prime}_{\theta,0}(x)\leq\theta v(v-1)-\beta^{\prime}(x)<0,

and so $H_{\theta,0}$ is strictly concave. Since $H^{\prime}_{\theta,0}(0)=0$ , $x=0$ is the unique global maximizer of $H_{\theta,0}(.)$ .

•

There exists $\theta_{2c}\in(0,\infty)$ such that for $\theta>\theta_{2c}$ , $0$ is not a global maximizer of $H_{\theta,0}(.)$
We consider two separate cases:
- –
  
  $\mu$ is stochastically non-negative.
  
  In this case there exists $x_{0}>0$ such that $\gamma(\beta(x_{0}))\in(0,\infty)$ . Then setting $\theta_{2c}:=x_{0}^{-v}\gamma(\beta(x_{0}))\in(0,\infty)$ , for $\theta>\theta_{2c}$ we have
  
  $H_{\theta,0}\left(x_{0}\right)=\theta x_{0}^{-v}-\gamma\left(\beta\left(x_{0}\right)\right)>0=H_{\theta,0}(0),$
  
  and so $0$ cannot be a global optimizer of $H_{\theta,0}(.)$
- –
  
  $v$ is even.
  
  If there exists $x_{0}>0$ such that $\gamma(\beta(x_{0}))\in(0,\infty)$ , we are through by previous argument. Othewise, since $\mu$ is not degenerate at $0$ , there exists $x_{0}<0$ such that $\gamma(\beta(x_{0}))\in(0,\infty)$ . Again setting $\theta_{2c}:=x_{0}^{-v}\gamma(\beta(x_{0}))\in(0,\infty)>0$ with $v$ even, the same proof works.

•

For any $\theta>0$ , let $x_{\theta}$ be any non-negative global optimizer of $H_{\theta,0}(.)$ . Then the map $\theta\mapsto x_{\theta}$ is non-decreasing.
Suppose by way of contradiction there exists $0<\theta_{1}<\theta_{2}<\infty$ such that $0\leq x_{\theta_{2}}<x_{\theta_{1}}$ . By optimality of $x_{\theta_{1}}$ we have

$\displaystyle\theta_{1}x^{v}_{\theta_{1}}-\gamma(\beta(x_{1}))$	$\displaystyle\geq$	$\displaystyle\theta_{1}x^{v}_{\theta_{2}}-\gamma(\beta(x_{2}))$
$\displaystyle\implies\theta_{1}(x^{v}_{\theta_{1}}-x^{v}_{\theta_{2}})$	$\displaystyle\geq$	$\displaystyle\gamma(\beta(x_{\theta_{1}}))-\gamma(\beta(x_{\theta_{2}}))$
$\displaystyle\implies\theta_{2}(x^{v}_{\theta_{1}}-x^{v}_{\theta_{2}})$	$\displaystyle>$	$\displaystyle\gamma(\beta(x_{\theta_{1}}))-\gamma(\beta(x_{\theta_{2}})).$

Here the last implication uses the fact $x_{\theta_{1}}>x_{\theta_{2}}\geq 0$ . But this contradicts the fact that $x_{\theta_{2}}$ is a global maximizer for $H_{\theta_{2},0}(.)$ .

Combining the last three claims, the conclusion of part (iii)(b) follows on setting $\theta_{c}:=\sup_{\theta>0}\{0\text{ is a global maximizer of }H_{\theta,0}(.)\}.$

∎

4. Proof of Main Lemmas

Proof of 2.5.

As before, we also choose $\tilde{p}\in(1,p)$ and $\tilde{q}\in(1,q)$ such that $\frac{1}{\tilde{p}}+\frac{1}{\tilde{q}}=1$ . Note that $\Upsilon(\cdot,\cdot)$ is well-defined on $\mathcal{R}$ (see 2.1) by using 2.1 (ii), with $W(\cdot,\cdot)$ as is, $\phi(x_{1},\ldots,x_{v})=\prod_{a=2}^{v}x_{a}$ , and $p,q$ replaced with $\tilde{p},\tilde{q}$ .

(i) Recall the definition of $\vartheta_{W,\nu}(\cdot)$ from (1.18), and the connection $\Upsilon(W,\nu):=\mathrm{Law}\left(U_{1},{\vartheta_{W,\nu}(U_{1})}\right)$ between $\Upsilon$ and $\vartheta_{W,\nu}$ from 2.1. We will prove the following stronger claim.

(4.1)

\sup_{W:\ \lVert W\rVert_{q\Delta}\leq C}\sup_{\nu:\ \mathfrak{m}_{p}(\nu)\leq C}\int_{0}^{1}|\vartheta_{W,\nu}(u)-\vartheta_{W,\nu_{L}}(u)|\,du\to 0,\quad\mbox{as}\ L\to\infty.

Towards this end, fix $L>1$ and note that

	$\displaystyle\;\;\;\;\|\vartheta_{W,\nu}(u)-\vartheta_{W,\nu_{L}}(u)\|$
	$\displaystyle\leq v\sum_{\begin{subarray}{c}A\subseteq\{2,\ldots,v\},\\ \|A\|\geq 1\end{subarray}}\mathbb{E}\left[\|\mathrm{Sym}[W](u,U_{2},\ldots,U_{v})\|\left(\prod_{a\in A}\|V_{a}\|1\{\|V_{a}\|>L\}\right)\left(\prod_{a\in A^{c}}\|V_{a}\|1\{\|V_{a}\|\leq L\}\right)\right].$

For every non empty fixed set $A\subseteq\{2,\ldots,v\}$ , an application of 2.1 part (ii) with $W(\cdot,\cdot)$ as is,

\phi(x_{1},\ldots,x_{v})=\left(\prod_{a\in A}|x_{a}|\mathbbm{1}(|x_{a}|\geq L)\right)\left(\prod_{a\in A^{c}}|x_{a}|\mathbbm{1}(|x_{a}|\leq L)\right),

and $p,q$ replaced by $\tilde{p}$ , $\tilde{q}$ on the above bound, gives

	$\displaystyle\;\;\;\;\sup_{W:\ \lVert W\rVert_{q\Delta}\leq C}\sup_{\nu:\ \mathfrak{m}_{p}(\nu)\leq C}\int_{0}^{1}\|\vartheta_{W,\nu}(u)-\vartheta_{W,\nu^{(L)}}(u)\|\,du$
	$\displaystyle\leq v\sup_{W:\ \lVert W\rVert_{q\Delta}\leq C}\sup_{\nu:\ \mathfrak{m}_{p}(\nu)\leq C}\sum_{A\subseteq\{2,\ldots,v\},\ \|A\|\geq 1}\lVert W\rVert_{\tilde{q}\Delta}^{\|E(H)\|}\Bigg{(}\left(\prod_{a\in A}\mathbb{E}_{\nu}[\|V_{a}\|^{\tilde{p}}\mathbbm{1}(\|V_{a}\|\geq L)]\right)$
	$\displaystyle\;\;\;\;\;\;\qquad\left(\prod_{a\in A^{c}}\mathbb{E}_{\nu}[\|V_{a}\|^{\tilde{p}}\mathbbm{1}(\|V_{a}\|\leq L)]\right)\Bigg{)}^{\frac{1}{\tilde{p}}}$
(4.2)		$\displaystyle\leq v2^{v}\sup_{W:\ \lVert W\rVert_{q\Delta}\leq C}\sup_{\nu:\ \mathfrak{m}_{p}(\nu)\leq C}L^{\tilde{p}-p}\lVert W\rVert_{\tilde{q}\Delta}^{\|E(H)\|}(1+\mathfrak{m}_{p}(\nu))^{\frac{v-1}{p}}\to 0,$

as $L\to\infty$ . This proves (4.1), and hence completes part (i).

(ii) Given $W\in\mathcal{W}$ , $\nu\in\mathcal{M}$ and any $u\in[0,1]$ , define

(4.3)

\mathfrak{R}(u;W):=\mathbb{E}[\mathrm{Sym}[|W|](u,U_{2},\ldots,U_{v})]

where $U_{2},\ldots,U_{v}\overset{i.i.d.}{\sim}\mathrm{Unif}[0,1]$ . For $k<\infty$ , and $T>0$ , define

c_{k}^{(T)}(u):=1\{\mathfrak{R}(u;W_{k})\leq T,\ \mathfrak{R}(u;W_{\infty})\leq T\},

for $u_{1}\in[0,1]$ . With this notation, by a truncation followed by a simple method of moments argument, the conclusion in part (ii) will follow if we can show the following:

(4.4)

\lim\limits_{T\to\infty}\limsup_{k\to\infty}\sup_{\nu\in\mathcal{M}^{(L)}}\int_{0}^{1}|\vartheta_{W_{k},\nu}(u)(1-c_{k}^{(T)}(u))|\,du=0,

(4.5)

\lim\limits_{T\to\infty}\limsup_{k\to\infty}\sup_{\nu\in\mathcal{M}^{(L)}}\int_{0}^{1}|\vartheta_{W_{\infty},\nu}(u)(1-c_{k}^{(T)}(u))|\,du=0,

(4.6)

\displaystyle\sup_{\nu\in\mathcal{M}^{(L)}}\Bigg{|}\int_{0}^{1}\big{(}\vartheta_{W_{k},\nu}(u)c_{k}^{(T)}(u)\big{)}^{r}\,du-\int_{0}^{1}\big{(}\vartheta_{W_{\infty},\nu}(u)c_{k}^{(T)}(u)\big{)}^{r}\,du\Bigg{|}\to 0,

as $k\to\infty$ , for every $T>0$ , and every $r\in\mathbb{N}$ .

Proof of (4.4). To begin, for any $W\in\mathcal{W}$ we have the bound

\displaystyle\sup_{\nu\in\mathcal{M}^{(L)}}|\vartheta_{W,\nu}(u)|\leq L^{v-1}\mathfrak{R}(u;W),

which gives

|\vartheta_{W_{k},\nu}(u)(1-c_{k}^{(T)}(u))|\leq L^{v-1}\mathfrak{R}(u;W_{k})\left(1\{\mathfrak{R}(u;W_{k})>T\}+1\{\mathfrak{R}(u;W_{\infty})>T\}\right).

Therefore, (4.4) will follow if we can show that

(4.7)

\lim_{T\to\infty}\limsup_{k\to\infty}\int_{0}^{1}\mathfrak{R}(u;W_{k})\left(1\{\mathfrak{R}(u;W_{k})>T\}+1\{\mathfrak{R}(u;W_{\infty})>T\}\right)\,du=0.

We now complete the proof based on the following claim, whose proof we defer.

(4.8)

\displaystyle\limsup_{k\to\infty}\int_{0}^{1}\mathfrak{R}^{q}(u;W_{k})\,du<\infty,\quad\int_{0}^{1}\mathfrak{R}^{q}(u;W_{\infty})\,du<\infty.

We will now deal with (4.7) term by term. First note that:

\displaystyle\int_{0}^{1}\mathfrak{R}(u;W_{k})1\{\mathfrak{R}(u;W_{k})>T\}\,du\leq\frac{1}{T^{q-1}}\int_{0}^{1}\mathfrak{R}^{q}(u;W_{k})\,du.

By the first claim in (4.8), the right hand side above converges to $0$ by taking $k\to\infty$ followed by $T\to\infty$ , thus proving the first claim in (4.7). For the second claim in (4.7), setting $\tilde{p}=q/(q-1)$ Hölder’s inequality gives

	$\displaystyle\;\;\;\;\int_{0}^{1}\mathfrak{R}(u;W_{k})1\{\mathfrak{R}(u;W_{\infty})>T\}\,du$
	$\displaystyle\leq\left(\int_{0}^{1}\mathfrak{R}^{q}(u;W_{k})\,du\right)^{\frac{1}{q}}\left(\int_{0}^{1}1\{\mathfrak{R}(u;W_{\infty})>T\}\,du\right)^{\frac{1}{\tilde{p}}}$
	$\displaystyle\leq\left(\int_{0}^{1}\mathfrak{R}^{q}(u;W_{k})\,du\right)^{\frac{1}{q}}\frac{1}{T^{\frac{1}{p}}}\left(\int_{0}^{1}\mathfrak{R}(u;W_{\infty})\,du\right)^{\frac{1}{\tilde{p}}},$

where the final quantity above converges to $0$ taking $k\rightarrow\infty$ followed by $T\rightarrow\infty$ using both claims in (4.8). This proves the second claim in (4.7), and hence completes the verification of (4.4), subject to proving (4.8).

Proof of (4.8). Note that

\displaystyle\mathfrak{R}^{q}(u;W_{k})

\displaystyle=\left(\mathbb{E}[\mathrm{Sym}[|W_{k}|](u,U_{2},\ldots,U_{v})]\right)^{q}\leq\mathbb{E}[\mathrm{Sym}[|W_{k}|^{q}](u,U_{2},\ldots,U_{v})],

where the inequality follows from Lyapunov’s inequality (the function $r\mapsto\mathbb{E}[|X|^{r}]^{1/r}$ is non-decreasing on $(0,\infty)$ ). On integrating over $u$ we get

\int_{0}^{1}\mathfrak{R}^{q}(u;W_{k})\,du\leq\mathbb{E}[\mathrm{Sym}[|W_{k}|^{q}](U_{1},\ldots,U_{v})]\leq\lVert W_{k}\rVert_{q\Delta}^{q},

where the last inequality follows from 2.1, part (iii). By our assumption $\limsup_{k\to\infty}\lVert W_{k}\rVert_{q\Delta}<\infty$ , the first conclusion in (4.8) follows. The second conclusion follows similarly.

Proof of (4.5). This follows the exact same line of argument as the proof of (4.4), and hence is omitted for brevity.

Proof of (4.6). Set $h_{\nu}(u):=\mathbb{E}_{\nu}[V|U=u]$ , and use the definition $\vartheta_{W_{k},\nu}(\cdot)$ in (1.18) to note that

	$\displaystyle\;\;\;\;\int_{0}^{1}\left(\vartheta_{W_{k},\nu}(u_{1})c_{k}^{(T)}(u_{1})\right)^{r}\,du_{1}$
	$\displaystyle=\int_{0}^{1}c_{k}^{(T)}(u_{1})\left(\int_{[0,1]^{(v-1)r}}\prod_{i=1}^{r}\left(\mathrm{Sym}[W_{k}](u_{1},u_{2}^{(i)},\ldots,u^{(i)}_{v})\prod_{a=2}^{v}h_{\nu}(u_{a}^{(i)})\,du_{a}^{(i)}\right)\right)\,du_{1}$

We can similarly write out an expression for $\int_{0}^{1}\left(\vartheta_{W_{\infty},\nu}(u_{1})c_{k}^{(T)}(u_{1})\right)^{r}\,du_{1}$ with $\mathrm{Sym}[W_{k}]$ is replaced by $\mathrm{Sym}[W_{\infty}]$ . Accordingly, to establish (4.6), replacing each $\mathrm{Sym}[W_{k}](u_{1},u_{2}^{(i)},\ldots,u_{v}^{(i)})$ by $\mathrm{Sym}[W_{\infty}](u_{1},u_{2}^{(i)},\ldots,u_{v}^{(i)})$ sequentially, it suffices to show that:

(4.9)

\lim_{k\to\infty}\sup_{\nu\in\widetilde{\mathcal{M}}^{(L)}}\ \big{|}\mathfrak{F}_{k}^{\nu,A}\big{|}=0,

for every fixed $L>0$ and $A\subseteq\{2,\ldots,r\}$ , where

	$\displaystyle\mathfrak{F}_{k}^{\nu,A}:=\int_{0}^{1}\bigg{(}\int_{[0,1]^{v-1}}(\mathrm{Sym}[W_{k}](u_{1},u_{2}^{(1)},\ldots,u_{v}^{(1)})-\mathrm{Sym}[W_{\infty}](u_{1},u_{2}^{(1)},\ldots,u_{v}^{(1)}))c_{k}^{(T)}(u_{1})$
	$\displaystyle\prod_{a=2}^{v}h_{\nu}(u_{a}^{(1)})\,du_{a}^{(1)}\bigg{)}\left(\int_{[0,1]^{\|A\|\times(v-1)}}\prod_{i\in A}\left(\mathrm{Sym}[W_{k}](u_{1},u^{(i)}_{2},\ldots,u^{(i)}_{v})c_{k}^{(T)}(u_{1})\prod_{a=2}^{v}h_{\nu}(u^{(i)}_{a})\prod_{a=2}^{v}\,du^{(i)}_{a}\right)\right)$
	$\displaystyle\left(\int_{[0,1]^{\|A^{c}\|\times(v-1)}}\prod_{i\in A^{c}}\left(\mathrm{Sym}[W_{\infty}](u_{1},u^{(i)}_{2},\ldots,u^{(i)}_{v})c_{k}^{(T)}(u_{1})\prod_{a=2}^{v}h_{\nu}(u^{(i)}_{a})\prod_{a=2}^{v}\,du^{(i)}_{a}\right)\right)c_{k}^{(T)}(u_{1})\,du_{1}.$

In order to establish (4.9), let us further define

\mathfrak{n}_{k}^{\nu,(T)}(u):=\int_{[0,1]^{v-1}}\mathrm{Sym}[W_{k}](u,u_{2},\ldots,u_{v})c_{k}^{(T)}(u)\prod_{a=2}^{v}h_{\nu}(u_{a})\prod_{a=2}^{v}\,du_{a},

\mathfrak{p}_{k}^{\nu,(T)}(u):=\int_{[0,1]^{v-1}}\mathrm{Sym}[W_{\infty}](u,u_{2},\ldots,u_{v})c_{k}^{(T)}(u)\prod_{a=2}^{v}h_{\nu}(u_{a})\prod_{a=2}^{v}\,du_{a},

and note that

(4.10)

\displaystyle\sup_{\nu\in\widetilde{\mathcal{M}}^{(L)}}\sup_{k\geq 1}\max\left\{\lVert\mathfrak{n}_{k}^{\nu,(T)}\rVert_{\infty},\lVert\mathfrak{p}_{k}^{\nu,(T)}\rVert_{\infty}\right\}\leq L^{v-1}T.

Proceeding to show (4.9), integrating with respect to all the variables other than $u_{1},u^{(i_{0})}_{2},\ldots,u^{(i_{0})}_{v}$ , we get

	$\displaystyle\big{\|}\mathfrak{F}_{k}^{\nu,A}\big{\|}$	$\displaystyle=\bigg{\|}\int_{0}^{1}\Bigg{(}\int_{[0,1]^{(v-1)}}\Bigg{(}\big{(}\mathrm{Sym}[W_{k}](u_{1},u_{2}^{(1)},\ldots,u_{v}^{(1)})-\mathrm{Sym}[W_{\infty}](u_{1},u_{2}^{(1)},\ldots,u_{v}^{(1)})\big{)}$
		$\displaystyle\quad c_{k}^{(T)}(u_{1})\prod_{a=2}^{v}h_{\nu}(u_{a}^{(1)})\prod_{a=2}^{v}\,du_{a}\Bigg{)}\Bigg{)}\left(\mathfrak{n}_{k}^{\nu,(T)}(u_{1})\right)^{\|A\|}\left(\mathfrak{p}_{k}^{\nu,(T)}(u_{1})\right)^{\|A^{c}\|}\,du_{1}\bigg{\|}$
		$\displaystyle\leq\frac{1}{v!}\sum_{\sigma\in S_{v}}\Bigg{\|}\int\left(\prod_{(a,b)\in E(H)}W_{k}(u_{\sigma(a)},u_{\sigma(b)})-\prod_{(a,b)\in E(H)}W_{\infty}(u_{\sigma(a)},u_{\sigma(b)})\right)$
		$\displaystyle\left(\prod_{a=2}^{v}h_{\nu}(u_{a})\right)c_{k}^{(T)}(u_{1})\left(\mathfrak{n}_{k}^{\nu,(T)}(u_{1})\right)^{\|A\|}\left(\mathfrak{p}_{k}^{\nu,(T)}(u_{1})\right)^{\|A^{c}\|}\prod_{a=1}^{v}\,du_{a}\Bigg{\|}.$

Observe that $|h_{\nu}|$ ’s are bounded by $L$ for $\nu\in\widetilde{\mathcal{M}}^{(L)}$ , $c_{k}^{(T)}$ is bounded by definition, and further $\mathfrak{n}_{k}^{\nu,(T)}$ and $\mathfrak{p}_{k}^{\nu,(T)}$ are both bounded by (4.10). The conclusion in (4.9) then follows from [6, Proposition 3.1 part (ii)].

(iii) Note that there exists a sequence of bounded continuous functions $W_{m}\in\mathcal{W}^{+}$ such that $\lVert W_{m}-W\rVert_{q}\to 0$ as $m\to\infty$ . The triangle inequality implies that given any $m\geq 1$ , $k\geq 1$ , we have:

	$\displaystyle d_{\ell}(\Upsilon(W,\nu_{k}),\Upsilon(W,\nu_{\infty}))$	$\displaystyle\leq d_{\ell}(\Upsilon(W,\nu_{k}),\Upsilon(W_{m},\nu_{k}))+d_{\ell}(\Upsilon(W_{m},\nu_{k}),\Upsilon(W_{m},\nu_{\infty}))$
(4.11)			$\displaystyle+d_{\ell}(\Upsilon(W,\nu_{\infty}),\Upsilon(W_{m},\nu_{\infty}))$

By part (ii), we have:

\lim_{m\to\infty}\sup_{k\in[1,\infty]}d_{\ell}(\Upsilon(W_{m},\nu_{k}),\Upsilon(W,\nu_{k}))=0.

Further from the definition of weak convergence we have, for every fixed $m$ ,

d_{\ell}(\Upsilon(W_{m},\nu_{k}),\Upsilon(W_{m},\nu_{\infty}))\to 0,\quad\mbox{as}\ k\to\infty.

Combining the two displays above with (4) establishes part (iii). ∎

Proof of 2.9.

Recall from (2.31) that $\widetilde{\mathfrak{B}}_{\theta}=G_{1}(\mathfrak{B}_{\theta}^{*})$ , where $G_{1}(x,y)=(x,\alpha^{\prime}(y))$ with $\alpha^{\prime}(.)$ continuous (see 1.3 for definition of $\alpha(.)$ ). The facts that $\mathfrak{B}_{\theta}^{*}\subseteq\widetilde{M}_{q}$ and $\widetilde{\mathfrak{B}}_{\theta}\subseteq\widetilde{M}_{p}$ follow directly from (1.25). It thus suffices to prove compactness of $\mathfrak{B}_{\theta}^{*}$ (which will imply compactness of $\widetilde{\mathfrak{B}}_{\theta}$ ).

To this effect, invoking (1.25) there exists $C>0$ such that $\Xi(F_{\theta})\in\widetilde{\mathcal{M}}_{p,C}$ (see (2.24) for the definition of $\widetilde{\mathcal{M}}_{p,C}$ ). Also by 2.6 the function $\Upsilon(W,\cdot)$ is continuous on $\Xi(F_{\theta})$ with respect to weak topology. Since $\Xi(F_{\theta})$ is compact in the weak topology (see 1.1 part (iv)), and $\mathfrak{B}_{\theta}^{*}=\Upsilon(W,\Xi(F_{\theta}))$ (from (2.13)), compactness of $\mathfrak{B}_{\theta}^{*}$ follows.

∎

Proof of 2.10.

(i) For any $L>0$ under $\mathbb{R}_{n,\theta}^{(1)}$ we have

(4.12)

\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left|d_{i}\left(X_{i}-X_{i}^{(L)}\right)\right|\leq\frac{D}{nL^{p-1}}\sum_{i=1}^{n}\mathbb{E}|X_{i}|^{p},

where $X_{i}^{(L)}:=X_{i}1\{|X_{i}|\leq L\}$ and $\|{\bf d}\|_{\infty}\leq D$ . The RHS of (4.12) converges to $0$ as $n\to\infty$ followed by $L\to\infty$ by using (1.22). Since $\alpha^{\prime}(\theta m_{i})=\mathbb{E}[X_{i}|X_{j},\ j\in[n],\ j\neq i]$ , setting

\mathcal{J}_{i}^{(L)}:=\mathbb{E}\left[X_{i}^{(L)}|X_{j},\ j\neq i\right],

we note that

\displaystyle\bigg{|}\alpha^{\prime}(\theta m_{i})-\mathcal{J}_{i}^{(L)}\bigg{|}\leq\mathbb{E}\left[|X_{i}|1(|X_{i}|>L)|X_{j},\ j\neq i\right]\leq\frac{1}{L^{p-1}}\mathbb{E}\left[|X_{i}|^{p}|X_{j},\ j\neq i\right].

Consequently,

(4.13)

\displaystyle\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\bigg{|}d_{i}\left(\alpha^{\prime}(\theta m_{i})-\mathcal{J}_{i}^{(L)}\right)\bigg{|}\leq\frac{D}{nL^{p-1}}\sum_{i=1}^{n}\mathbb{E}|X_{i}|^{p},

which converges to $0$ as $n\to\infty$ followed by $L\to\infty$ , by using (1.22) and the fact that $p>1$ . Combining (4.12) and (4.13), it suffices to show $\sum_{i=1}^{n}d_{i}\left(X_{i}^{(L)}-\mathcal{J}_{i}^{(L)}\right)=o_{P}(1)$ . Towards this direction, we further define, for $i\neq j$ ,

\mathcal{J}^{(L)}_{i,j}:=\mathbb{E}\left[X_{i}^{(L)}|X_{k},\ k\neq\{i,j\},\ X_{j}=0\right],

and observe that

	$\displaystyle\;\;\;\;\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}d_{i}\Big{(}X_{i}^{(L)}-\mathcal{J}_{i}^{(L)}\Big{)}\right]^{2}$
	$\displaystyle=\frac{1}{n^{2}}\sum_{i=1}^{n}d_{i}^{2}\mathbb{E}\left(X_{i}^{(L)}-\mathcal{J}_{i}^{(L)}\right)^{2}+\frac{1}{n^{2}}\sum_{i\neq j}d_{i}d_{j}\mathbb{E}\left[\left((X_{i}^{(L)}-\mathcal{J}_{i}^{(L)}\right)\left(X_{j}^{(L)}-\mathcal{J}_{j}^{(L)}\right)\right]$
	$\displaystyle\leq\frac{4D^{2}L^{2}}{n}+\frac{1}{n^{2}}\sum_{i\neq j}d_{i}d_{j}\mathbb{E}\left[\left(X_{i}-\mathcal{J}_{i,j}^{(L)}+\mathcal{J}_{i,j}^{(L)}-\mathcal{J}_{i}^{(L)}\right)\left(X_{j}^{(L)}-\mathcal{J}_{j}^{(L)}\right)\right].$

For $i\neq j$ the random variable $X_{i}^{(L)}-\mathcal{J}_{i,j}^{(L)}$ is measurable with respect to the sigma field generated by $\{X_{k},\ k\in[n],\ k\neq j\}$ , and consequently,

\mathbb{E}\left[\left(X_{i}^{(L)}-\mathcal{J}_{i,j}^{(L)}\right)\left(X_{j}^{(L)}-\mathcal{J}_{j}^{(L)}\right)\right]=0,

for $i\neq j$ . Combining the last two displays gives

(4.14)

\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}d_{i}\Big{(}X_{i}^{(L)}-\mathcal{J}_{i}^{(L)}\Big{)}\right]^{2}\leq\frac{4D^{2}L^{2}}{n}+\frac{2LD^{2}}{n^{2}}\sum_{i\neq j}\mathbb{E}\left|\mathcal{J}_{i,j}^{(L)}-\mathcal{J}_{i}^{(L)}\right|.

It suffices to show that the second term in the RHS of (4.14) converges to $0$ for every fixed $D,L$ . To control this second term, define

(4.15)

m_{i,j}:=\frac{v}{n^{v-1}}\sum_{\begin{subarray}{c}(k_{2},\ldots,k_{v})\\ \in\mathcal{S}(n,v,\{i,j\})\end{subarray}}\mathrm{Sym}[Q_{n}](i,k_{2},\ldots,k_{v})\left(\prod_{m=2}^{v}X_{k_{m}}\right)

for $i\neq j$ , where $\mathcal{S}(n,v,\{i,j\})$ denotes the set of all distinct tuples in $[n]^{v-1}$ , such that none of the elements equal to $i$ or $j$ . For any $K>0$ , by the triangle inequality we have the following for any $i\neq j$ ,

	$\displaystyle\;\;\;\;\;\frac{1}{n^{2}}\sum_{i\neq j}\mathbb{E}\left\|\mathcal{J}_{i,j}^{(L)}-\mathcal{J}_{i}^{(L)}\right\|$
	$\displaystyle\leq\frac{1}{n^{2}}\sum_{i\neq j}\mathbb{E}\left[\left(\left\|\mathcal{J}_{i,j}^{(L)}\right\|+\left\|\mathcal{J}_{i}^{(L)}\right\|\right)\left(\mathbbm{1}(\|m_{i,j}\|\geq K)+\mathbbm{1}(\|m_{i}\|\geq K)\right)\right]$
	$\displaystyle\;\;\;\;\;\;\;\;\;\;+\frac{1}{n^{2}}\sum_{i=1}^{n}\mathbb{E}\left[\left\|\mathcal{J}_{i,j}^{(L)}-\mathcal{J}_{i}^{(L)}\right\|\mathbbm{1}(\|m_{i,j}\|\leq K,\ \|m_{i}\|\leq K)\right]$
(4.16)		$\displaystyle\leq\frac{2L}{n^{2}K}\sum_{i\neq j}\mathbb{E}\left(\|m_{i,j}\|+\|m_{i}\|\right)+\frac{1}{n^{2}}\sum_{i\neq j}^{n}\mathbb{E}\left[\left\|\mathcal{J}_{i,j}^{(L)}-\mathcal{J}_{i}^{(L)}\right\|\mathbbm{1}(\|m_{i,j}\|\leq K,\ \|m_{i}\|\leq K)\right].$

It suffices to show that the RHS of (4) converges to $0$ as $n\to\infty$ , followed by $K\to\infty$ . Now let us complete this proof based on the following claim, whose proof we defer:

(4.17)

\sum_{i\neq j}\mathbb{E}|m_{i}-m_{i,j}|=O(n).

By combining (4.17) with (1.23), we also have:

(4.18)

\sum_{i\neq j}\mathbb{E}|m_{i,j}|=O(n^{2}).

By combining (4.18) with (1.23), it is immediate that the first term in the RHS of (4) converges to $0$ as $n\to\infty$ , followed by $K\to\infty$ . For the second term in the RHS of (4), let us define the function:

\mathfrak{E}_{L}(t):=\frac{\int_{|x|\leq L}x\exp(tx)\,d\mu(x)}{\int_{-\infty}^{\infty}\exp(tx)\,d\mu(x)}=\mathbb{E}_{X\sim\mu_{t}}[X\mathbbm{1}(|X|\leq L)],

where $\mu_{t}$ is the exponential tilt of $\mu$ as introduced in 1.3. From standard properties of exponential families, $\mathfrak{E}_{L}(\cdot)$ has a continuous derivative on $\mathbb{R}$ and therefore,

\sup_{|t|\leq|\theta|K}|\mathfrak{E}_{L}^{\prime}(t)|\leq\mathfrak{c},

where $\mathfrak{c}>0$ depends on $|\theta|$ , $L$ , and $K$ . Hence,

	$\displaystyle\;\;\;\;\;\frac{1}{n^{2}}\sum_{i\neq j}\mathbb{E}\left[\left\|\mathcal{J}_{i,j}^{(L)}-\mathcal{J}_{i}^{(L)}\right\|\mathbbm{1}(\|m_{i,j}\|\leq K,\ \|m_{i}\|\leq K)\right]$
	$\displaystyle=\frac{1}{n^{2}}\sum_{i\neq j}\mathbb{E}\left[\left\|\mathfrak{E}(\theta m_{i,j})-\mathfrak{E}(\theta m_{i})\right\|\mathbbm{1}(\|m_{i,j}\|\leq K,\ \|m_{i}\|\leq K)\right]$
(4.19)		$\displaystyle\leq\frac{\mathfrak{c}\|\theta\|}{n^{2}}\sum_{i\neq j}\mathbb{E}\|m_{i,j}-m_{i}\|=O\left(\frac{1}{n}\right),$

for every fixed $\theta$ , $L$ , and $K$ . This completes the proof that (4) converges to $0$ as $n\to\infty$ followed by $K\to\infty$ .

Proof of (4.17). The symmetry of $\mathrm{Sym}[Q_{n}]$ implies

\displaystyle|m_{i}-m_{i,j}|\leq\frac{v}{n^{v-1}}|X_{j}|\sum_{\begin{subarray}{c}(k_{3},\ldots,k_{v})\\ \in\mathcal{S}(n,v-1,\{i,j\})\end{subarray}}\mathrm{Sym}[|Q_{n}|](i,j,k_{3},\ldots,k_{v})\left(\prod_{m=3}^{v}|X_{k_{m}}|\right).

Using this, we bound the left hand side of (4) below.

	$\displaystyle\frac{1}{n}\sum_{i\neq j}\left\|m_{i}-m_{i,j}\right\|$	$\displaystyle\leq\frac{v}{n^{v}}\sum_{i\neq j}\sum_{\begin{subarray}{c}(k_{3},\ldots,k_{v})\\ \in\mathcal{S}(n,v,\{i,j\})\end{subarray}}\mathrm{Sym}[\|Q_{n}\|](i,j,k_{3},\ldots,k_{v})\|X_{j}\|\left(\prod_{m=3}^{v}\|X_{k_{m}}\|\right)$
		$\displaystyle\leq v\left(\frac{1}{n^{v}}\sum_{(k_{1},\ldots,k_{v})\in[n]^{v}}\|\mathrm{Sym}[\|Q_{n}\|](k_{1},\ldots,k_{v})\|^{q}\right)^{\frac{1}{q}}\left(\frac{1}{n}\sum_{i=1}^{n}\|X_{i}\|^{p}\right)^{\frac{v-1}{p}}$
		$\displaystyle\leq v\lVert W_{Q_{n}}\rVert_{q\Delta}\left(\frac{1}{n}\sum_{i=1}^{n}\|X_{i}\|^{p}\right)^{\frac{v-1}{p}},$

where the second inequality follows from Hölder’s inequality, and the third inequality uses 2.1 part (c). The above display, on taking expectation, gives

\displaystyle\frac{1}{n}\sum_{i\neq j}\mathbb{E}|m_{i}-m_{i,j}|\leq v\lVert W_{Q_{n}}\rVert_{q\Delta}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}|X_{i}|^{p}\right)^{\frac{v-1}{p}}.

Here, we have used Lyapunov’s inequality coupled with the observation that $v-1\leq p$ . As $\limsup_{n\to\infty}\lVert W_{Q_{n}}\rVert_{q\Delta}<\infty$ by (1.8), an application of (1.22) in the last display above completes the proof of (4.17). ∎

Proof of 2.10.

(ii) Choose $\tilde{q}<q$ and $\tilde{p}<p$ such that $\tilde{p}^{-1}+\tilde{q}^{-1}=1$ . Fixing $L>0$ we have

\frac{1}{n}\bigg{|}\sum_{i=1}^{n}m_{i}(X_{i}-X_{i}^{(L)})\bigg{|}\leq\frac{1}{L^{\frac{p}{\tilde{p}}-1}}\left(\frac{1}{n}\sum_{i=1}^{n}|m_{i}|^{\tilde{q}}\right)^{\frac{1}{\tilde{q}}}\left(\frac{1}{n}\sum_{i=1}^{n}|X_{i}|^{p}\right)^{\frac{1}{\tilde{p}}}=o_{p}(1),

where the limit is to be understood as $n\to\infty$ followed by $L\to\infty$ . Here we used (1.22) and (1.23). Now, from standard analysis we have the existence of a $C_{1}$ function $\psi_{L}:\mathbb{R}\to\mathbb{R}$ such that $\psi_{L}(x)=x$ for $|x|\leq L$ , $|\psi_{L}(x)|\leq|x|$ , $\lVert\psi_{L}\rVert_{\infty}<\infty$ , and $\lVert\psi_{L}^{\prime}\rVert_{\infty}<\infty$ . This gives

|m_{i}-\psi_{L}(m_{i})|^{\tilde{q}}\leq 2^{\tilde{q}}|m_{i}|^{\tilde{q}}{1}\{|m_{i}|>L\}\leq\frac{2^{\tilde{q}}}{L^{q-\tilde{q}}}|m_{i}|^{q}.

Using this bound along with Hölder’s inequality, we get:

	$\displaystyle\frac{1}{n}\bigg{\|}\sum_{i=1}^{n}(m_{i}-\psi_{L}(m_{i}))X_{i}^{(L)}\bigg{\|}$	$\displaystyle\leq\left(\frac{1}{n}\sum_{i=1}^{n}\|m_{i}-\psi_{L}(m_{i})\|^{\tilde{q}}\right)^{\frac{1}{\tilde{q}}}\left(\frac{1}{n}\sum_{i=1}^{n}\|X_{i}\|^{\tilde{p}}\right)^{\frac{1}{\tilde{p}}}$
		$\displaystyle\leq\frac{2}{L^{\frac{q}{\tilde{q}}-1}}\left(\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|^{q}\right)^{\frac{1}{\tilde{q}}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{\tilde{p}}\right)^{\frac{1}{\tilde{p}}}=o_{P}(1)$

as $n\to\infty$ followed by $L\to\infty$ , on using (1.22) and (1.23). Combining the above displays we get

\displaystyle\frac{1}{n}\bigg{|}\sum_{i=1}^{n}m_{i}X_{i}-\sum_{i=1}^{n}\psi_{L}(m_{i})X_{i}^{(L)}\bigg{|}=o_{p}(1),

as $n\to\infty$ followed by $L\to\infty$ . A similar computation shows

\displaystyle\frac{1}{n}\bigg{|}\sum_{i=1}^{n}m_{i}\mathbb{E}[X_{i}|X_{j},\ j\neq i]-\sum_{i=1}^{n}\psi_{L}(m_{i})\mathbb{E}[X_{i}^{(L)}|X_{j},\ j\neq i]\bigg{|}=o_{p}(1)

in the same sense. Using the last two displays above, it suffices to show that

(4.20)

\displaystyle\frac{1}{n}\bigg{|}\sum_{i=1}^{n}\psi_{L}(m_{i})\left(X_{i}^{(L)}-\mathbb{E}[X_{i}^{(L)}|X_{j},\ j\neq i]\right)\bigg{|}=o_{p}(1),

as $n\to\infty$ for fixed $L$ . Towards this direction, we will use the definitions of $m_{i,j}$ , $\mathcal{J}_{i}^{(L)}$ , $\mathcal{J}_{i,j}^{(L)}$ from the proof of 2.10 part (a). Observe that

	$\displaystyle\;\;\;\;\;\frac{1}{n^{2}}\mathbb{E}\bigg{\|}\sum_{i=1}^{n}\psi_{L}(m_{i})\left(X_{i}^{(L)}-\mathcal{J}_{i}^{(L)}\right)\bigg{\|}^{2}$
	$\displaystyle\leq\frac{L^{2}\lVert\psi_{L}\rVert_{\infty}^{2}}{n}+\frac{1}{n^{2}}\sum_{i\neq k}\mathbb{E}\left[\psi_{L}(m_{i})\psi_{L}(m_{k})(X_{i}^{(L)}-\mathcal{J}_{i}^{(L)})(X_{k}^{(L)}-\mathcal{J}_{k}^{(L)})\right].$

By Markov’s inequality, in order to establish (4.20), it suffices to show that the above display converges to $0$ as $n\to\infty$ for any fixed $L$ . As

\mathbb{E}\left[\psi_{L}(m_{i,k})\psi_{L}(m_{k})(X_{i}^{(L)}-\mathcal{J}_{i,k}^{(L)})(X_{k}^{(L)}-\mathcal{J}_{k}^{(L)})\right]=0

for $i\neq k$ , it suffices to show that

\frac{1}{n^{2}}\sum_{i\neq k}\mathbb{E}|\mathcal{J}_{i}^{(L)}-\mathcal{J}_{i,k}^{(L)}|=o(1),\qquad\frac{1}{n^{2}}\sum_{i\neq k}\mathbb{E}|\psi_{L}(m_{i})-\psi_{L}(m_{i,k})|=o(1).

The left hand display is what we bounded in (4). As $|\psi_{L}(m_{i})-\psi_{L}(m_{i,k})|\leq\lVert\psi_{L}^{\prime}\rVert_{\infty}|m_{i}-m_{i,k}|$ , the right hand display above follows directly from (4.17). ∎

5. Appendix

In this Section, we will prove the auxiliary lemmas from earlier in the paper. Section 5.1 collects all results on the properties of the base measure $\mu$ , and Section 5.2 contains some general probabilistic convergence results.

5.1. Proofs of Lemmas 1.7, 2.2, and 2.3

Proof of 1.7.

With $\mathfrak{v}_{\theta,B,\mu}(x)=\theta x^{2}+Bx-x\beta(x)+\alpha(\beta(x))$ as in the statement of the lemma, differentiation gives $\mathfrak{v}^{\prime}_{\theta,B,\mu}(x)=2\theta x+B-\beta(x)$ . Using 2.3 part (ii) we get $\lim_{x\to\pm\infty}\mathfrak{v}^{\prime}_{\theta,B,\mu}(x)=\pm\infty$ (since $p\geq 2)$ , and so the continuous function $\mathfrak{v}_{\theta,B,\mu}(.)$ attains its global maximizers on $\mathbb{R}$ , and any maximizer (local or global) satisfies $\mathfrak{v}^{\prime}_{\theta,B,\mu}(x)=2\theta x+B-\beta(x)=0$ , which is equivalent to solving $\tilde{\mathfrak{v}}_{\theta,B,\mu}(x)=0$ , where

(5.1)

\tilde{\mathfrak{v}}_{\theta,B,\mu}(x):=x-\alpha^{\prime}(2\theta x+B),\quad\tilde{\mathfrak{v}}^{\prime}_{\theta,B,\mu}(x):=1-2\theta\alpha^{\prime\prime}(2\theta x+B).

(i) Here $B=0$ , and symmetry of $\mu$ gives $\alpha^{\prime}(0)=\tilde{\mathfrak{v}}_{\theta,0,\mu}(0)=0$ . To show that $0$ is the only root of $\tilde{\mathfrak{v}}_{\theta,0,\mu}(\cdot)$ (and hence the unique maximizer of $\mathfrak{v}_{\theta,0,\mu}$ ), using symmetry of $\mu$ it suffices to show that $\tilde{\mathfrak{v}}_{\theta,0,\mu}$ does not have any other roots on $(0,\infty).$ To this effect, using (1.29) it follows that $\alpha^{\prime\prime}(.)$ is non-increasing on $(0,\infty)$ , and so $\tilde{\mathfrak{v}}_{\theta,0,\mu}$ is convex using (5.1). Since $\tilde{\mathfrak{v}}^{\prime}_{\theta,0,\mu}(0)=0$ , it follows that $0$ is also a global minimizer of $\tilde{\mathfrak{v}}_{\theta,0,\mu}(\cdot)$ , and so $\tilde{\mathfrak{v}}_{\theta,0,\mu}$ is non-positive. If there exists a positive root $x_{0}$ of $\tilde{\mathfrak{v}}_{\theta,0,\mu}(\cdot)$ , then by convexity (and symmetry) we have $\tilde{\mathfrak{v}}_{\theta,0,\mu}\equiv 0$ on $[-x_{0},x_{0}]$ . But this implies $\alpha^{\prime}(\cdot)$ is linear on this domain, and so $\alpha(\cdot)$ must be a quadratic, which is only possible only if $\mu$ is a Gaussian. This contradicts (1.6), and hence completes the proof of part (i).

(ii) By symmetry, it suffices to consider the case $B>0$ . Comparing $x$ with $-x$ and using the symmetry of $\mu$ , it follows that all global maximizers lie in $[0,\infty)$ . Also in this case $\alpha^{\prime}(B)>0$ , which implies $\tilde{\mathfrak{v}}_{\theta,B,\mu}(0)<0$ . As $\lim\limits_{x\to\infty}\tilde{\mathfrak{v}}_{\theta,B,\mu}(x)=\infty$ by 2.3 part (i), $\tilde{\mathfrak{v}}_{\theta,B,\mu}(\cdot)$ either has a unique positive root, or at least $3$ positive roots. If the latter holds, using (5.1) $\alpha^{\prime\prime}(\cdot)$ must have two positive roots $(x_{1},x_{2})$ , which on using (1.29) gives that $\alpha^{\prime\prime\prime}(\cdot)\equiv 0$ on the interval $[x_{1},x_{2}]$ . As in part (i), this implies that $\mu$ is Gaussian, a contradiction to (1.6). Thus $\mathfrak{v}(\cdot)$ has a unique positive maximizer $t_{\theta,B,\mu}$ .

(iii) In this case $\tilde{\mathfrak{v}}_{\theta,B,\mu}(0)=0$ and $\tilde{\mathfrak{v}}_{\theta,B,\mu}^{\prime}(0)<0$ . Therefore, $\tilde{\mathfrak{v}}_{\theta,B,\mu}(\cdot)$ either has a unique positive root or at least $3$ positive roots. From there we argue, similar to part (ii) above, that $\tilde{\mathfrak{v}}_{\theta,0,\mu}(\cdot)$ has exactly one positive root $t_{\theta,0,\mu}$ . By symmetry, it follows that $-t_{\theta,0,\mu}$ is the unique negative root of $\tilde{\mathfrak{v}}_{\theta,0,\mu}(\cdot)$ , and $\pm t_{\theta,0,\mu}(\cdot)$ are the global maximizers of $\mathfrak{v}_{\theta,0,\mu}$ . ∎

Proof of 2.2.

The function $\beta(.)$ is smooth ( $C^{\infty}$ ) on $\mathcal{N}$ , and the function $\gamma(.)$ is smooth on $\mathbb{R}$ . Consequently, the function $\gamma(\beta(.))$ is smooth on $\mathcal{N}$ . To verify continuity on ${\rm cl}(\mathcal{N})$ , it suffices to cover the (possible) boundary cases:

•

If $a:=\sup\{\mathcal{N}\}<\infty$ , then $\lim_{u\to a}\gamma(\beta(u))=\gamma(\beta(a))=\gamma(\infty)$ .
•

If $b:=\inf\{\mathcal{N}\}>-\infty$ , then $\lim_{u\to b}\gamma(\beta(u))=\gamma(\beta(b))=\gamma(-\infty)$ .

We will only prove the first case, as the other case follows similarly. Note that,

\lim_{u\to a}\beta(u)=\infty\Rightarrow\lim_{u\to a}\mu_{\beta(u)}=\delta_{a},

where the second limit is in weak topology. Further,

(5.2)

\liminf_{u\to a}\gamma(\beta(u))=\liminf_{u\to\sup\{\mathcal{N}\}}D(\mu_{\beta(u)}|\mu)\geq D(\delta_{a}|\mu)=\gamma(\infty)

by the lower semi-continuity of Kullback-Leibler divergence. If $\mu(\{a\})=0$ , then $\gamma(\infty)=\infty$ , and (5.2) yields the desired conclusion. If $\mu(\{a\})>0$ , then $\gamma(\infty)=-\log{\mu(\{a\})}$ . Also, for any $\theta\in\mathbb{R}$ , we have

\alpha(\theta)=\log{\int\exp(\theta x)\,d\mu(x)}\geq\theta a+\log{\mu(\{a\})}.

For all $u$ such that $\beta(u)>0$ (which holds for all $u$ close to $a$ ), this gives

\gamma(\beta(u))=u\beta(u)-\alpha(\beta(u))\leq u\beta(u)-a\beta(u)-\log{\mu(\{a\})}\leq\log{\mu(\{a\})}=\gamma(\infty).

Combining the above display with (5.2) gives $\lim_{u\to a}\gamma(\beta(u))=\gamma(\infty),$ as desired. ∎

Proof of 2.3.

(i) We prove $\lim_{\theta\to\infty}\frac{\alpha^{\prime}(\theta)}{\theta^{\frac{1}{p-1}}}=0$ , noting that the proof of the other limit is similar. To this effect, we consider the following two cases separately:

•

$\mu(0,\infty)>0$ .

Fixing $\theta>0$ and $\delta>0$ , we have:

	$\displaystyle\frac{\|\alpha^{\prime}(\theta)\|}{\theta^{\frac{1}{p-1}}}$	$\displaystyle\leq\frac{\int_{\mathbb{R}}\|y\|\exp(\theta y)\,d\mu(y)}{\theta^{\frac{1}{p-1}}\int_{\mathbb{R}}\exp(\theta y)\,d\mu(y)}$
		$\displaystyle\leq\frac{\delta\theta^{\frac{1}{p-1}}\int_{\|y\|\leq\delta\theta^{\frac{1}{p-1}}}\exp(\theta y)\,d\mu(y)+\int_{\|y\|\geq\delta\theta^{\frac{1}{p-1}}}\|y\|\exp(\theta y)\,d\mu(y)}{\theta^{\frac{1}{p-1}}\int\exp(\theta y)\,d\mu(y)}$
		$\displaystyle\leq\delta+\frac{\int_{\mathbb{R}}\|y\|\exp(\|y\|^{p}\delta^{1-p})d\mu(y)}{\theta^{\frac{1}{p-1}}\int_{\mathbb{R}}\exp(\theta y)d\mu(y)},$

where we use the bound $|\theta y|\leq|y|^{p}\delta^{1-p}$ on the set $|y|\geq\delta|\theta|^{\frac{1}{p-1}}$ . Letting $\theta\to\infty$ we have $\int_{\mathbb{R}}\exp(\theta y)d\mu(y)\to\infty$ , as $\mu(0,\infty)>0$ . Since the numerator in the second term in the display above is finite invoking (1.6), the second term above converges to $0$ as $\theta\to\infty$ , allowing us to conclude $\limsup_{\theta\to\infty}\frac{|\alpha^{\prime}(\theta)|}{\theta^{\frac{1}{p-1}}}\leq\delta$ . Since $\delta>0$ is arbitrary, the desired limit follows.

•

$\mu(0,\infty)=0$ .
In this case, $\alpha^{\prime}(\theta)\leq 0$ . Since $\alpha^{\prime}(\cdot)$ is non-decreasing, $\lim_{\theta\to\infty}\alpha^{\prime}(\theta)$ exists as a finite (non-positive) number. Consequently we have $\lim_{\theta\to\infty}\frac{\alpha^{\prime}(\theta)}{\theta^{\frac{1}{p-1}}}=0$ .

(ii) We only study the case when $x\to\sup\{\mathcal{N}\}$ . If $\sup\{\mathcal{N}\}<\infty$ , then the conclusion is immediate as the denominator converges to a finite number while the numerator diverges. Therefore, we only focus on the case $\sup\{\mathcal{N}\}=\infty$ . To this effect, fixing $M>0$ using part (i) gives that for all $x$ large enough (depending on $M$ ) we have

\alpha^{\prime}(Mx)\leq x^{\frac{1}{p-1}}\Leftrightarrow Mx\leq\beta\left(x^{\frac{1}{p-1}}\right).

Taking limit gives

\liminf_{x\to\infty}\frac{\beta\left(x^{\frac{1}{p-1}}\right)}{x}\geq M.

Since $M$ is arbitrary, we conclude the desired conclusion follows. ∎

5.2. Proofs of 2.1 and Lemmas 2.7 and 2.8

Proof of 2.1.

(i) and (ii) These are direct consequences of [10, Proposition 2.19] and [6, Lemma 2.2].

(iii) With $\mathrm{Sym}[|W|]$ as in 1.6, we have

	$\displaystyle\mathbb{E}\left[\mathrm{Sym}[\|W\|](U_{1},\ldots,U_{v})\right]^{q}$	$\displaystyle=\mathbb{E}\bigg{\|}\frac{1}{v!}\sum_{\sigma\in\mathcal{S}_{v}}\prod_{(a,b)\in E(H)}\|W\|(U_{\sigma(a)},U_{\sigma(b)})\bigg{\|}^{q}$
		$\displaystyle\leq\frac{1}{v!}\sum_{\sigma\in\mathcal{S}_{v}}\mathbb{E}\prod_{(a,b)\in E(H)}\|W\|^{q}(U_{\sigma(a)},U_{\sigma(b)})\leq\lVert W\rVert_{q\Delta}^{q\|E(H)\|},$

where the first inequality uses Lyapunov’s inequality, and the second inequality follows from 2.1 part (ii), with $W$ replaced by $|W|^{q}$ . ∎

Proof of 2.7.

By using (2.25), it follows that the sequence $\{\xi_{n}\}_{n\geq 1}$ is tight. Passing to a subsequence, w.l.o.g. we can assume $\xi_{n}\stackrel{{\scriptstyle d}}{{\to}}\xi_{\infty}$ , where $\mathbb{P}(\xi_{\infty}\in\mathcal{F})=1$ (as $\mathcal{F}$ is closed). By the Portmanteau Theorem,

(5.3)

\displaystyle\mathbb{P}(\xi_{\infty}\in K^{c})\leq\limsup\limits_{n\to\infty}\mathbb{P}(\xi_{n}\in K^{c})=0.

Next we will show that $g(\xi_{n})\stackrel{{\scriptstyle d}}{{\to}}g(\xi_{\infty})$ . Towards this direction let $H\subseteq g(\mathcal{F})$ be a closed set. We will write $g^{-1}(H)$ to denote the inverse image of the set $H$ under $g$ . Another application of the Portmanteau Theorem implies:

	$\displaystyle\limsup\limits_{n\to\infty}\mathbb{P}(g(\xi_{n})\in H,\ \xi_{n}\in K)=$	$\displaystyle\limsup\limits_{n\to\infty}\mathbb{P}(\xi_{n}\in g^{-1}(H)\cap K)$
(5.4)		$\displaystyle\geq$	$\displaystyle\mathbb{P}(\xi_{\infty}\in g^{-1}(H)\cap K).$

The last line uses the fact that $g^{-1}(H)\cap K$ is closed which in turn follows from the continuity of $g$ on $K$ . Finally, by (2.25) and (5.3), we have:

(5.5)

\displaystyle\limsup\limits_{n\to\infty}|\mathbb{P}(g(\xi_{n})\in H,\ \xi_{n}\in K)-\mathbb{P}(g(\xi_{n})\in H)|=0,

(5.6)

\displaystyle\mathbb{P}(\xi_{\infty}\in g^{-1}(H)\cap K)=\mathbb{P}(\xi_{\infty}\in g^{-1}(H)).

By combining (5.2), (5.5), and (5.6), it follows that

\limsup\limits_{n\to\infty}P(g(\xi_{n})\in H)\geq\mathbb{P}(g(\xi_{\infty})\in H).

By the Portmanteau theorem, this yields $g(\xi_{n})\stackrel{{\scriptstyle d}}{{\to}}g(\xi_{\infty})$ . So for any $\varepsilon>0$ we get

\displaystyle\limsup_{n\to\infty}\mathbb{P}(d_{Y}(g(\xi_{n}),g(\mathcal{F}))\geq\varepsilon)\leq\mathbb{P}(d_{Y}(g(\xi_{\infty}),g(\mathcal{F}))\geq\varepsilon)=0

as $g(\xi_{\infty})\in g(\mathcal{F})$ a.s. Since $\varepsilon>0$ is arbitrary, $d_{Y}(g(\xi_{\infty}),g(\mathcal{F}))\stackrel{{\scriptstyle P}}{{\to}}0$ , as desired. ∎

Proof of 2.8.

(i) Since $\limsup_{n\to\infty}\mathbb{E}|f_{n}(U)|^{p}<\infty$ , it follows that the sequence $\{|f_{n}(U)|^{p^{\prime}}\}_{n\geq 1}$ is uniformly integrable, and so $\mathbb{E}|f_{\infty}(U)|^{p^{\prime}}<\infty$ . By standard approximation results, given any $\varepsilon>0$ , there exists $h:[0,1]\to\mathbb{R}$ (depending on $\varepsilon$ ) such that $h$ is continuous on $[0,1]$ and $\mathbb{E}|h(U)-f_{\infty}(U)|^{p^{\prime}}<\varepsilon$ . Continuous mapping theorem gives $f_{n}(U)-h(U)\overset{D}{\longrightarrow}f_{\infty}(U)-h(U)$ . Since $|f_{n}(U)-f_{\infty}(U)|^{p^{\prime}}$ is uniformly integrable, $\lVert f_{n}-f_{\infty}\rVert_{p^{\prime}}\longrightarrow\lVert f_{\infty}-h\rVert_{p^{\prime}}\leq\varepsilon$ . As $\varepsilon>0$ is arbitrary, this completes the proof of part (a).

(ii) The conclusion follows by applying part (a) on the sequence of measures alternating between $(U,f(U))$ and $(U,g(U))$ along odd and even subsequences. ∎

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(1.22)	$\displaystyle\limsup\limits_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}\|X_{i}\|^{p}\geq C\right)<0$	$\displaystyle\Rightarrow\sum_{i=1}^{n}\mathbb{E}\|X_{i}\|^{p}=O(n),$
(1.23)	$\displaystyle\limsup\limits_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|^{q}\geq C\right)<0$	$\displaystyle\Rightarrow\sum_{i=1}^{n}\mathbb{E}\|m_{i}\|^{q}=O(n),$
(1.24)	$\displaystyle\limsup\limits_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})\|^{p}\geq C\right)<0$	$\displaystyle\Rightarrow\sum_{i=1}^{n}\mathbb{E}\|\alpha^{\prime}(\theta m_{i})\|^{p}=O(n).$

	$\displaystyle\|m_{i}\|$	$\displaystyle\leq\frac{v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|\prod_{a=2}^{v}\|X_{i_{a}}\|$
		$\displaystyle\leq v\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{1}{q}}\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\prod_{a=2}^{v}\|X_{i_{a}}\|^{p}\right)^{\frac{1}{p}}$
(2.4)			$\displaystyle=v\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{1}{q}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p}}.$

	$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\|m_{i}\|^{q}$	$\displaystyle\leq v^{q}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{q(v-1)}{p}}\left(\frac{1}{n^{v}}\sum_{(i_{1},\ldots,i_{v})\in[n]^{v}}\|\mathrm{Sym}[Q_{n}](i_{1},i_{2},\ldots,i_{v})\|^{q}\right)$
(2.5)			$\displaystyle\leq v^{q}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{q(v-1)}{p}}\\|W_{Q_{n}}\\|_{q\Delta}^{q\|E(H)\|},$

	$\displaystyle\|\alpha^{\prime}(\theta m_{i})\|$	$\displaystyle=\bigg{\|}\alpha^{\prime}\left(\frac{\theta v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in\mathcal{S}(n,v,i)}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|\prod_{a=2}^{v}\|X_{i_{a}}\|\right)\bigg{\|}$
		$\displaystyle\leq c_{\mu}\left(\frac{\theta v}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|\prod_{a=2}^{v}\|X_{i_{a}}\|\right)^{\frac{1}{p-1}}$
		$\displaystyle\leq c_{\mu}(\theta v)^{\frac{1}{p-1}}\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})\in[n]^{v-1}}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{1}{q(p-1)}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p(p-1)}}.$

	$\displaystyle\;\;\;\;\;\frac{1}{n}\sum_{i=1}^{n}\|\alpha^{\prime}(\theta m_{i})\|^{p}$
	$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}c_{\mu}^{p}(\theta v)^{\frac{p}{p-1}}\left(\frac{1}{n^{v-1}}\sum_{(i_{2},\ldots,i_{v})}\|\mathrm{Sym}[Q_{n}](i,i_{2},\ldots,i_{v})\|^{q}\right)^{\frac{p}{q(p-1)}}\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p-1}}$
(2.7)		$\displaystyle\leq c_{\mu}^{p}(\theta v)^{\frac{p}{p-1}}\left(1+\lVert W_{Q_{n}}\rVert_{q\Delta}^{q\|E(H)\|}\right)\left(\frac{1}{n}\sum_{j=1}^{n}\|X_{j}\|^{p}\right)^{\frac{v-1}{p-1}}.$

Gibbs measures with multilinear forms

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

1.1. Main results

Definition 1.1.

Definition 1.2.

Definition 1.3.

Definition 1.4.

Definition 1.5.

Proposition 1.1.

Remark 1.1.

1.1.1. Replica-symmetry

Definition 1.6.

Definition 1.7.

Theorem 1.2 (Replica-symmetry).

Remark 1.2.

Example 1.1 (Necessity of conditions on μ\mu).

Example 1.2 (Necessity of θ>0\theta>0).

1.1.2. Weak laws and tail bounds

Definition 1.8 (Local fields).

Theorem 1.3.

Remark 1.3.

Theorem 1.4.

Corollary 1.5.

Remark 1.4.

Remark 1.5.

Theorem 1.6.

Remark 1.6.

1.2. Examples

1.2.1. Quadratic interaction models with symmetric base measure

Lemma 1.7.

Remark 1.7.

Proposition 1.8.

Remark 1.8.

1.2.2. Gibbs measures with higher order interactions

Theorem 1.9.

1.3. Proof overview and future scope

1.4. Outline of the paper

2. Proof of Main Results

2.1. Proofs of 1.1 and 1.2

Proposition 2.1.

Remark 2.1.

Proof of 1.1.

Remark 2.2.

Lemma 2.2.

Proof of 1.2.

Lemma 2.3.

Proof of 1.4.

2.2. Proof of 1.3

Remark 2.3.

Definition 2.1.

Lemma 2.4.

Proof of (2.15).

Lemma 2.5.

Proof of (2.16).

Proof of (2.17).

Corollary 2.6.

Lemma 2.7.

Proof of (2.18).

2.3. Proofs of 1.5 and 1.6

Lemma 2.8.

Lemma 2.9.

Proof of 1.5.

Lemma 2.10.

Proof of 1.6.

3. Proof of Results from Section 1.2

Proof of 1.8.

Proof of 1.9.

4. Proof of Main Lemmas

Proof of 2.5.

Proof of 2.9.

Proof of 2.10.

Proof of 2.10.

5. Appendix

5.1. Proofs of Lemmas 1.7, 2.2, and 2.3

Proof of 1.7.

Proof of 2.2.

Proof of 2.3.

Example 1.1 (Necessity of conditions on $\mu$ ).

Example 1.2 (Necessity of $\theta>0$ ).