On the rate of convergence of the martingale central limit theorem in Wasserstein distances

Xiaoqin Guo The work of XG is supported by Simons Foundation through Collaboration Grant for Mathematicians #852943. Department of Mathematical Sciences, University of Cincinnati , 2815 Commons Way, Cincinnati, OH 45221, USA

Abstract

For martingales with a wide range of integrability, we will quantify the rate of convergence of the central limit theorem via Wasserstein distances of order $r$ , $1\leq r\leq 3$ . Our bounds are in terms of Lyapunov’s coefficients and the $\mathscr{L}^{r/2}$ fluctuation of the total conditional variances. We will show that our Wasserstein-1 bound is optimal up to a multiplicative constant.

1 Introduction

In this paper, we consider the rate of convergence of the martingale central limit theorem (CLT) under Wasserstein distances. Let $Y_{\cdot}=\{Y_{m}\}_{m=1}^{n}$ be a square integrable martingale difference sequence (mds) with respect to $\sigma$ -fields $\{\mathcal{F}_{m}\}_{m=0}^{n}$ . Here $\mathcal{F}_{0}$ denotes the trivial $\sigma$ -field. For $1\leq m\leq n$ , let $s_{m}^{2}=\sum_{i=1}^{m}E[Y_{i}^{2}]$ and

$\displaystyle\sigma_{m}^{2}(Y)$	$\displaystyle=E[Y_{m}^{2}\|\mathcal{F}_{m-1}],$
$\displaystyle V_{n}^{2}$	$\displaystyle=\frac{1}{s_{n}^{2}}\sum_{i=1}^{n}\sigma_{i}^{2}(Y),$
$\displaystyle X_{m}$	$\displaystyle=\frac{1}{s_{n}}\sum_{i=1}^{m}Y_{i}.$	(1)

For a mds $\{Y_{i}\}_{i=1}^{\infty}$ , when $\lim_{n\to\infty}V_{n}^{2}=1$ in probability and Lindeberg’s condition is satisfied, it is well-known that

X_{n}\Rightarrow\mathcal{N}\quad\text{ as }n\to\infty,

where $\mathcal{N}\sim\mathcal{N}(0,1)$ denotes a standard normal random variable.

To quantify such a convergence in distribution, one of the most important metrics is the Wasserstein distance which has deep roots in the optimal transport theory [42]. Recall that, for two probability measures $\mu,\nu$ on $\mathbb{R}$ , their Wasserstein distance (also called minimal distance) $\mathcal{W}_{r}$ of order $r$ , $r\geq 1$ , is defined by

\mathcal{W}_{r}(\mu,\nu)=\inf\{\lVert U-V\rVert_{r}:U\sim\mu,V\sim\nu\},

where $\lVert U\rVert_{r}=E[|U|^{r}]^{1/r}$ . With abuse of notations, we may use random variables as synonyms of their distributions. E.g., for $U\sim\mu,V\sim\nu$ , we may write $\mathcal{W}_{r}(\mu,\nu)$ as $\mathcal{W}_{r}(U,V)$ .

When $r=1$ , recall that $\mathcal{W}_{1}$ admits the following alternative representations:

\mathcal{W}_{1}(U,V)=\sup_{h\in\rm{Lip}_{1}}\left\{E[h(U)]-E[h(V)]\right\}=\int_{\mathbb{R}}\Bigr{\lvert}F_{U}(x)-F_{V}(x)\Bigl{\rvert}\mathrm{d}x

where $\rm{Lip}_{1}$ denotes the set of all 1-Lipschitz functions, and $F_{U}(x):=P(U\leq x)$ .

Throughout the paper, we use $c,C$ to denote positive constants which may change from line to line. Unless otherwise stated, $C_{p},c_{p}$ denote constants depending only on the parameter $p$ . We write $A\lesssim B$ if $A\leq CB$ , and $A\asymp B$ if $A\lesssim B$ and $A\gtrsim B$ . We also use notations $A\lesssim_{p}B$ , $A\asymp_{p}B$ to indicate that the multiplicative constant depends on the parameter $p$ .

1.1 Earlier results in the literature

There is an immense literature on the convergence rate of the CLT for independent random variables. Such results are often phrased in terms of Lyapunov’s coefficient (i.e., the term in the $\mathscr{L}^{p}$ Lyapunov condition). To be specific, for $p\geq 1$ , set¹¹1Note that the meaning of our $L_{p}$ notation is slightly different from that of [2, 27, 25, 38, 4] in the literature whose $L_{p}$ means the $L_{p}^{p}$ in our paper.

L_{p}=L_{p}(Y):=\frac{1}{s_{n}}\left(\sum_{i=1}^{n}E[|Y_{i}|^{p}]\right)^{1/p}.

(2)

Note that typically $L_{p}$ is of size $O(n^{1/p-1/2})$ .

When $\{Y_{i}\}_{i=1}^{n}$ is a centered independent sequence, a nonuniform estimate of Bikjalis [2] implies $\mathcal{W}_{1}(X_{n},\mathcal{N})\leq cL_{p}^{p}$ , $p\in(2,3]$ , extending the $\mathcal{W}_{1}$ bounds of Esseen [16], Zolotarev [45], and Ibragimov [29]. For $r>2$ , Sakhanenko [40] proved that $\mathcal{W}_{r}(X_{n},\mathcal{N})\leq cL_{r}$ which is optimal for independent $\mathscr{L}^{r}$ -integrable variables. For $1\leq r\leq 2$ , it is established by Rio [37, 38] that

\mathcal{W}_{r}(X_{n},\mathcal{N})\leq c_{r}L_{r+2}^{(r+2)/r}.

(3)

In particular, when $\{Y_{i}\}_{i=1}^{n}$ have roughly the same $\mathscr{L}^{r+2}$ moments, the $\mathcal{W}_{r}$ bound in (3) achieves the optimal rate $O(n^{-1/2})$ . Bobkov [4] confirmed Rio’s conjecture that (3) holds for all $r\geq 1$ . Cf. [38, 4] and references therein. For further developments, see e.g. [44, 20, 8, 22, 14, 6, 33] for work in the multivariate setting and e.g. [21, 33] for results on random variables with local-dependence.

It is natural to to ask whether the martingale CLT can be quantified by similar Wasserstein metric bounds in terms of $L_{p}$ . However, despite its theoretical importance, there are very few such results for (general) martingales compared to the independent case, let alone questions on optimal rates.

When $\{Y_{i}\}_{i=1}^{n}$ is a mds, the non-uniform bound of the distribution functions by Joos [31] (which generalizes Haeusler and Joos [26]) implies that

\mathcal{W}_{1}(X_{n},\mathcal{N})\leq c_{p,q}\left(L_{p}^{p/(1+p)}+\lVert V_{n}^{2}-1\rVert_{q}^{q/(2q+1)}\right)\quad\text{ for }2<p<\infty,q\geq 1.

(4)

In the case $M=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}<\infty$ , the nonuniform bound of Joos [30] implies

\mathcal{W}_{1}(X_{n},\mathcal{N})\leq c_{q}\left(\frac{M}{s_{n}}\log\big{(}e+\frac{s_{n}^{2}}{M^{2}}\big{)}+\lVert V_{n}^{2}-1\rVert_{q}^{q/(2q+1)}\right)\,\text{ for }q\geq 1.

(5)

Still for the $\mathscr{L}^{\infty}$ case, Dung, Son, Tien [12] improved the last term in (5) to be $C_{q}\lVert V^{2}-1\rVert_{q}^{1/2}$ for any $q>1/2$ . Under the condition

P(V_{n}^{2}=1)=1,

(6)

Röllin’s [39, (2.1)] result inferred that (See also [18, Lemma 2.1])

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim L_{3}.

(7)

Fan and Ma [18] dropped the condition (6) and obtained

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim_{q}L_{3}+\lVert V_{n}^{2}-1\rVert_{q}^{1/2}+\frac{1}{s_{n}}\max_{i=1}^{n}\lVert Y_{i}\rVert_{2q},\quad\forall q\geq 1.

(8)

Recently, assuming (6), Fan, Su [19, Corollary 2.5] implicitly extended (7) to be

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim L_{p}\quad\text{ for }2<p\leq 3.

(9)

Dedecker, Merlevède, and Rio [11] proved $\mathcal{W}_{r}$ bounds, $r\in(0,3]$ , that involve the quantities $\sum_{i=\ell}^{n}E[Y_{i}^{2}|\mathcal{F}_{\ell-1}]-E[Y_{i}^{2}]$ (instead of $V_{n}^{2}-1$ in our bounds), $2\leq\ell\leq n$ , which can be suitably bounded in many situations.

Readers may refer to [10, 9, 39, 19, 11] for Wasserstein bounds for the martingale CLT under other special conditions (e.g. the sequence being stationary, $\sigma_{i}$ ’s being close to be deterministic, $\sigma_{i}$ ’s are uniformly bounded from below, or certain variants of the $\mathscr{L}^{\infty}$ case, etc.).

Remark 1.

Unlike the Wasserstein metric, the Kolmogorov distance bounds for the martingale CLT has been thoroughly investigated since the 1970s. One of the earliest results is due to Heyde, Brown [28] which states that, for $2<p\leq 4$ ,

d_{K}(X_{n},\mathcal{N}):=\sup_{x\in\mathbb{R}}\lvert F_{X_{n}}(x)-F_{\mathcal{N}}(x)\rvert\lesssim_{p}L_{p}^{p/(p+1)}+\lVert V_{n}^{2}-1\rVert_{p/2}^{p/(2p+2)}.

(10)

When the mds is $\mathscr{L}^{\infty}$ , i.e., $\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}=M<\infty$ , Bolthausen [5] showed that

d_{K}(X_{n},\mathcal{N})\lesssim_{M}\frac{n\log n}{s_{n}^{3}}+\min\bigg{\{}\lVert V_{n}^{2}-1\rVert_{\infty}^{1/2},\lVert V_{n}^{2}-1\rVert_{1}^{1/3}\bigg{\}}

(11)

and that the first term $\tfrac{\log n}{n^{1/2}}$ is optimal for the case $s_{n}^{2}=n$ . Haeusler [24, 25] generalized (10) to $2<p<\infty$ and showed that the first term $L_{p}^{p/(p+1)}$ is exact. Joos [31] proved that the second term in (10) can be replaced by $\lVert V_{n}^{2}-1\rVert_{q}^{q/(2q+1)}$ , $q\geq 1$ . Mourrat [34] improved the second term in (11) to $(\lVert V_{n}^{2}-1\rVert_{q}+s_{n}^{-2})^{q/(2q+1)}$ , $q\geq 1$ . Cf. also [26, 31, 23, 35, 15, 17, 19] and references therein for work in this direction.

1.2 Motivation and our contributions

Our paper concerns the $\mathcal{W}_{r}$ convergence rates for the martingale CLT, $1\leq r\leq 3$ .

Let us comment on some weaknesses of the aforementioned $\mathcal{W}_{1}$ bounds.

When the martingale differences are at least $\mathscr{L}^{p}$ -integrable, $2<p\leq 3$ , the best $\mathcal{W}_{1}$ rates given by (7), (9), (8) are typically $n^{1/p-1/2}\geq n^{-1/6}$ , leaving a big gap between the rate $O(n^{-1/2}\log n)$ in the $\mathscr{L}^{\infty}$ case (5), not to mention that the condition (6) imposed in (9) is too restrictive for general martingales. Compared to (9), the exponent of $L_{p}$ in (4) is clearly not optimal, at least for $2<p\leq 3$ . But (4) does not assume (6), and it offers a typical $\mathcal{W}_{1}$ rate $O(n^{(2-p)/(2p+2)})$ which is better than $n^{-1/6}$ for $p>7/2$ . However, the constant $c_{p}$ in (4) is expected to grow linearly as $p\to\infty$ , rendering (4) a useless bound when $p$ is bigger than $n^{1/2}$ .

Notice that all of the results discussed above are $\mathcal{W}_{1}$ bounds. There are rarely any results on the $\mathcal{W}_{r}$ bounds, $r>1$ , in terms of $L_{p}$ for (general) martingales.

Can we obtain $\mathcal{W}_{r}$ bounds, $r\geq 1$ , in terms of the Lyapunov coefficient (2) for the martingale CLT that generalize all of the previous results (4), (5), (7), (8), (9)? Further, what are the optimal rates and how to justify their optimality?

Is it possible to get Wasserstein distance bounds for the CLT so that the constant coefficents do not blow up as the integrability of the martingale increases? Such estimates would be important when we have a limited sample size, and they allow us to exploit the integrability of the martingale to obtain better rates.

In theory and in applications, there are numerous stochastic processes that do not fit into any of the $\mathscr{L}^{p}$ category, $2<p<\infty$ , cf. [41, 32, 43]. Can we quantify the CLT for martingales with much wider spectrum of integrability than $\mathscr{L}^{p}$ , $p\ll n^{1/2}$ ?

Motivated by these questions, we will prove the following results.

(1)

We will obtain $\mathcal{W}_{r}$ , $1\leq r\leq 3$ , convergence rates for martingales which are Orlicz-integrable. For instance, if

$A:=\frac{1}{n}\sum_{i=1}^{n}\psi(|Y_{i}|)<\infty$ (12)

for appropriate convex function $\psi$ that grows at most polynomially fast, then

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim_{\psi}\frac{A\vee 1}{s_{n}}\psi^{-1}(n)+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2},$ (13)

where $\psi^{-1}$ denotes the inverse function of $\psi$ .

Our result greatly generalizes the known $\mathcal{W}_{1}$ bounds (4), (7), (8), (9) for $\mathscr{L}^{p}$ martingales, $2<p\leq 3$ . Moreover, we will prove that both terms $\tfrac{1}{s_{n}}\psi^{-1}(n)$ and $\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}$ in this $\mathcal{W}_{1}$ bound are optimal.

(2)

We will derive Wasserstein bounds whose constant coefficients do not depend on the integrability of the variables. Take the $\mathcal{W}_{1}$ distance for example, if (12) holds for $\psi$ in a wide class of convex functions, we show that

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\frac{A\vee 1}{s_{n}}\psi^{-1}(n)\log n+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.

(14)

This explains the presence of the $\log n$ term in (5), and it implies that if the mds is $\mathscr{L}^{\infty}$ bounded, i.e., $M=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}<\infty$ , then

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\frac{M}{s_{n}}\log(e+\frac{s_{n}^{2}}{M^{2}})+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.

The novelty of this result lies not only on the fact that it encompasses an even larger spectrum (all the way up to $\mathscr{L}^{\infty}$ ) of integrability than our first bound (13), but also that it yields a better bound than (13) when the order of the “integrability” $\psi$ is bigger than the logarithm of the sample size (i.e., beyond $O(\log n)$ ).

(3)

Similar bounds for the $\mathcal{W}_{r}$ distances in terms of the Lyapunov coefficient and $\lVert V_{n}^{2}-1\rVert_{r/2}$ , $1<r\leq 3$ , will be established as well.

1.3 Structure of the paper

The organization of our paper is as follows.

Subsection 1.4 contains definitions of N-functions and the corresponding Orlicz norm for random sequences. In Section 2, we present our main Wasserstein distance bounds (Theorems 5,7, and 8) and the optimality of the $\mathcal{W}_{1}$ rate (Proposition 10). In Section 3, using Taylor expansion and Lindeberg’s telescopic sum argument, we will derive $\mathcal{W}_{r}$ bounds in terms of the conditional moments for martingales with $V_{n}^{2}=1$ a.s.. As consequences, $\mathcal{W}_{1},\mathcal{W}_{2}$ bounds (Corollary 15) will be obtained for martingales that satisfy certain special conditions.

Section 4 is devoted to the proof of our main results. Our proof consists of the following components. First, we truncate the martingale as Haeusler [25] and elongate it as Dvoretzky [13] to turn it into a sequence with bounded increments and $V_{n}^{2}=1$ . We will bound the error of this modification in terms of the Lyapunov conefficient and $(V_{n}^{2}-1)$ . (See Section 4.1.) Then, as a crucial technical step, we use Young’s inequality within conditional expectations to “decouple” the Lyapunov coefficient and $\sigma_{i}^{2}(Y)$ ’s from the conditional moments, and turn the $\mathcal{W}_{r}$ bound to be an optimization problem over three parameters. This argument is robust enough for us to deal with martingales with very flexible integrability. Our another key observation is that there should be different bounds for the two different scenarios when the martingale is “at most $\mathscr{L}^{p}$ ” and “more integrable than $\mathscr{L}^{p}$ ”. By possibly sacrificing a small factor (e.g. $\log n$ ), we can make the constants of the bound independent of the integrability $\psi$ , which leads to bounds of type (14).

Finally, in Section 5 we construct examples to show that our $\mathcal{W}_{1}$ bound is optimal. We leave some open questions in Section 6.

1.4 Preliminaries: N-functions and an Orlicz-norm for sequences

To generalize the notion of $\mathcal{L}^{p}$ integrability, we recall the definitions of N-function and the corresponding Orlicz-norm in this subsection.

Definition 2.

A convex function $\psi:[0,\infty)\to[0,\infty)$ is called an N-function if it satisfies $\lim_{x\to 0}\psi(x)/x=0$ , $\lim_{x\to\infty}\psi(x)/x=\infty$ , and $\psi(x)>0$ for $x>0$ . For two functions $\psi_{1},\psi_{2}\in[0,\infty)^{[0,\infty)}$ , we write

\psi_{1}\preccurlyeq\psi_{2}\quad\text{ or }\quad\psi_{2}\succcurlyeq\psi_{1}

if $\frac{\psi_{2}(x)}{\psi_{1}(x)}$ is non-decreasing on $(0,\infty)$ .

Note that every N-function $\psi$ satisfies $\psi\succcurlyeq x$ . See [1].

Denote the Fenchel-Legendre transform of $\psi$ by $\psi_{*}\in[0,\infty)^{[0,\infty)}$ , i.e.,

\psi_{*}(x)=\sup\{xy-\psi(y):y\in\mathbb{R}\}\quad\text{ for }x\geq 0.

Then $\psi_{*}$ is still an N-function.Young’s inequality states that

xy\leq\psi(x)+\psi_{*}(y)\quad\text{ for all }x,y\geq 0.

(15)

Another useful relation between the pair $\psi,\psi_{*}$ is

x\leq\psi^{-1}(x)\psi_{*}^{-1}(x)\leq 2x,\quad\forall x\in[0,\infty).

(16)

See [1] for a proof and for more properties of N-functions.

Definition 3.

For any N-function $\psi$ and $n\in\mathbb{N}$ , the $\mathscr{L}^{\psi}$ -Orlicz norm of a random sequence $Y=\{Y_{m}\}_{m=1}^{n}$ with length $n$ is defined by

\lVert Y\rVert_{\psi}:=\inf\big{\{}c>0:\frac{1}{n}\sum_{i=1}^{n}E\left[\psi(|Y_{i}|/c)\right]\leq 1\big{\}}.

(17)

In particular, when $n=1$ , i.e., $Y=Y_{1}$ is a single random variable, we still write

\lVert Y_{1}\rVert_{\psi}=\inf\{c>0:E[\psi(|Y_{1}|/c)]\leq 1\}.

When $\psi(x)=x^{p}$ , $p\geq 1$ , we simply write $\lVert Y\rVert_{\psi}$ as $\lVert Y\rVert_{p}$ . Notice that

	$\displaystyle\lVert Y\rVert_{p}$	$\displaystyle=\left(\frac{1}{n}\sum_{i=1}^{n}E[\|Y_{i}\|^{p}]\right)^{1/p},$		(18)
	$\displaystyle\text{and }\quad\lim_{p\to\infty}\lVert Y\rVert_{p}$	$\displaystyle=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}=:\lVert Y\rVert_{\infty}.$		(19)

Remark 4.

(a)

Using the property $\psi(\lambda x)\geq\lambda\psi(x)$ for $\lambda\geq 1$ for N-functions, it is easily seen that the integrability condition (12) implies $\lVert Y\rVert_{\psi}\leq A\vee 1$ .
(b)

Since this paper concerns square-integrable martingales, we are only interested in sup-quadratic N-functions $\psi\succcurlyeq x^{2}$ . For instance, $x^{2}\log(x+1)$ , $x^{p}$ (with $p\geq 2$ ), $e^{x}-x-1$ , $\exp(x^{\beta})-1$ , $\exp\left(\ln(x+1)^{\beta}\right)-1$ (with $\beta>1$ ) are among such N-functions.
(c)

The meaning of our notation $\lVert Y\rVert_{p}$ is different from some work in the literature, e.g. [5, 19] where it means $\max_{i=1}^{n}\lVert Y_{i}\rVert_{p}$ . Note that $\lVert Y\rVert_{\psi}\leq\max_{i=1}^{n}\lVert Y_{i}\rVert_{\psi}$ .
(d)

Removing the $\tfrac{1}{n}$ in (18) only changes a multiplicative factor of $\lVert Y\rVert_{p}$ . However, for general $\lVert\cdot\rVert_{\psi}$ , the presence of $\tfrac{1}{n}$ in (17) is crucial for the definition–removing it would drastically change the meaning of the norm.

2 Main results

Recall $\lVert Y\rVert_{\infty}$ in (19). For the mds $\{Y_{i}\}_{i=1}^{n}$ and an N-function $\psi$ , we generalize the notation $L_{p}$ in (2) to be

L_{\psi}=\frac{1}{s_{n}}\lVert Y\rVert_{\psi}\psi^{-1}(n),\quad L_{\infty}=\frac{1}{s_{n}}\lVert Y\rVert_{\infty},

(20)

where $\psi^{-1}$ denotes the inverse function of $\psi$ .

Our first main results are two $\mathcal{W}_{1}$ bounds that generalize (4), (5), (7), (8), (9).

Theorem 5 ( $\mathcal{W}_{1}$ bounds).

Let $\{Y_{m}\}_{m=1}^{n}$ be a martingale difference sequence. Recall the notations $X_{n},V_{n}^{2},L_{\psi},L_{\infty},\preccurlyeq,\succcurlyeq$ in (1), (20), and Definition 2.

(i)

For any N-function $\psi\succcurlyeq x^{2}$ ,

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim L_{\psi}\log\left(e+L_{\psi}^{-2}\right)+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.$ (21)
(ii)

For any N-function $\psi$ with $x^{2}\preccurlyeq\psi\preccurlyeq x^{p}$ , $p>2$ ,

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim pL_{\psi}+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.$ (22)

As consequences, for any $p>2$ ,

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\min\left\{p,\,\log(e+L_{p}^{-2})\right\}L_{p}+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.

In particular, for the $\mathscr{L}^{\infty}$ case, by (19),

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim L_{\infty}\log\left(e+L_{\infty}^{-2}\right)+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}.

Remark 6.

(a)

For any sup-quadratic N-function $\psi\succcurlyeq x^{2}$ , by Lemma A.1,

$s_{n}=\lVert Y\rVert_{2}n^{1/2}\lesssim_{\psi(1)}\lVert Y\rVert_{\psi}n^{1/2}.$

Hence, by the definition of $L_{\psi}$ in (20),

$L_{\psi}\gtrsim_{\psi(1)}\psi^{-1}(n)n^{-1/2}\gtrsim_{\psi(1)}\psi^{-1}(1)n^{-1/2}.$

Thus, $\log(e+L_{\psi}^{-2})$ in Theorem 5(i) is dominated by $\log(n)$ .
(b)

Both bounds (21) and (22) in Theorem 5 have their strengths and weaknesses.

Apparently, the second bound (22) gives better rates for mds which are at most polynomially integrable. E.g., for the “barely more than square integrable” case $\psi=x^{2}\log(x+1)$ , (22) yields $\frac{1}{s_{n}}(\frac{n}{\log n})^{1/2}(\log\log n)\lVert Y\rVert_{\psi}$ as the first term in the bound. However, (22) becomes a trivial bound for $p\gg n^{1/2}$ .

Although the first bound (21) is seemingly $\log(n)$ factor worse than the latter, it is applicable to martingales with more general integrability. For instance, if the martingale is exponentially integrable, taking $\psi=e^{x}-x-1$ , Theorem 5(i) yields $\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\lVert Y\rVert_{\psi}(\log n)^{2}/s_{n}+\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}$ , where the first term is better than the $O(n^{1/p}/s_{n})$ rates offered by (22) for any $p>0$ .
(c)

In some sense, the term $\lVert V_{n}^{2}-1\rVert$ quantifies the extent of decorrelation of the process. For independent sequences, $(V_{n}^{2}-1)$ is $0$ . In general, $(V_{n}^{2}-1)$ could converge to $0$ arbitrarily fast, depending on how decorrelated the process is.
(d)

The term $(V_{n}^{2}-1)$ is essential for bounds of type

$\mathcal{W}_{r}(X_{n},\mathcal{N})\leq C_{1}L_{\psi}^{a}+C_{2}\lVert V_{n}^{2}-1\rVert_{c}^{b},$ (23)

i.e., we cannot allow $C_{2}$ to be $0$ . For example, let $B$ be such that $P(B=1\pm\tfrac{1}{2})=\tfrac{1}{2}$ and let $(\xi_{i})_{i=1}^{n}$ be i.i.d. $\mathcal{N}(0,1)$ variables. Define $Y_{i}=(B/n)^{1/2}\xi_{i}$ . Then $\lim_{n\to\infty}\mathcal{W}_{r}(\sum_{i=1}^{n}Y_{i},\mathcal{N})=\mathcal{W}_{r}(\sqrt{B}\xi_{1},\mathcal{N})\neq 0$ . Whereas, for $p>2$ , $L_{p}=O(n^{1/p-1/2})\to 0$ as $n\to\infty$ . Thus $C_{2}\neq 0$ .

Our next main results concern the $\mathcal{W}_{2}$ and $\mathcal{W}_{3}$ convergence rates.

Theorem 7 ( $\mathcal{W}_{2}$ bounds).

Let $\{Y_{m}\}_{m=1}^{n}$ be a martingale difference sequence.

(i)

For any N-function $\psi\preccurlyeq x^{3}$ such that $x\mapsto\psi(\sqrt{x})$ is convex,

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim L_{\psi}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}.$

(ii)

For any N-function $\psi\succcurlyeq x^{3}$ ,

\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim_{\psi(1)}L_{\psi}+\frac{\lVert Y\rVert_{\psi}}{s_{n}}\left[ng^{-1}(\psi^{-1}(n)^{2})\right]^{1/4}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2},

where $g$ denotes the inverse function of $x\mapsto\tfrac{\psi(x)}{x}$ .

As consequences, for $p\in(2,\infty)$ ,

\displaystyle\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\left\{\begin{array}[]{lr}L_{p}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}&\text{ when }p\in(2,3],\\ n^{1/4+1/(2p^{2}-2p)}s_{n}^{-1}\lVert Y\rVert_{p}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}&\text{ when }p>3.\end{array}\right.

In particular, taking $p\to\infty$ ,

\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\frac{n^{1/4}}{s_{n}}\lVert Y\rVert_{\infty}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}.

(24)

Theorem 8 ( $\mathcal{W}_{3}$ bound).

Let $\{Y_{m}\}_{m=1}^{n}$ be a martingale difference sequence. Then

\mathcal{W}_{3}(X_{n},\mathcal{N})\lesssim L_{3}+\lVert V_{n}^{2}-1\rVert_{3/2}^{1/2}.

Remark 9.

For the ease of our presentation, we only present results in terms of the $\mathcal{W}_{r}$ metrics, $r\in\{1,2,3\}$ . Readers can use the interpolation inequality

\mathcal{W}_{r}(X_{n},\mathcal{N})\leq\mathcal{W}_{j}(X_{n},\mathcal{N})^{j(k-r)/(k-j)}\mathcal{W}_{k}(X_{n},\mathcal{N})^{k(r-j)/(k-j)},\quad j<r<k,

to easily infer other $\mathcal{W}_{r}$ bounds, $r\in(1,3)$ .

The following proposition justifies that the terms $L_{\psi}$ and the exponent $1/2$ of $\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}$ within the $\mathcal{W}_{1}$ bound (22) are optimal.

Proposition 10.

(Optimality of the $\mathcal{W}_{1}$ rates).

(1)
For any $p>2$ , there exists a constant $N$ depending on $p$ such that for any $n>N$ , we can find a martingale $\{X_{i}\}_{i=1}^{n}$ which satisfies
- •
  
  $V_{n}^{2}=1$ , a.s.;
- •
  
  $L_{p}\lesssim_{p}(\log n)^{-1}$ ;
- •
  
  $\mathcal{W}_{1}(X_{n},\mathcal{N})\asymp_{p}L_{p}$ .
(2)
For any $n\geq 2$ , there exists a martingale $\{X_{i}\}_{i=1}^{n}$ that satisfies
- •
  
  $\lVert V_{n}^{2}-1\rVert_{1/2}\lesssim(\log n)^{-2}$ ;
- •
  
  $\mathcal{W}_{1}(X_{n},\mathcal{N})\asymp\lVert V_{n}^{2}-1\rVert_{1/2}^{1/2}$ .

3 Martingales with $V_{n}^{2}=1$

In this section we will derive $\mathcal{W}_{r}$ bounds, $r\in\{1,2,3\}$ , for martingales with $V_{n}^{2}=1$ . As in [39, 18, 11, 19], we will use Lindeberg’s argument and Taylor expansion to bound the Wasserstein distance with a sum involving the third order conditional moments. Our derivation of the $\mathcal{W}_{r}$ bounds for $r>1$ also relies on an observation of Rio [38] that relates the Wasserstein distances to Zolotarev’s ideal metric (25).

Note that [39, 18, 19] use Stein’s method to express the $\mathcal{W}_{1}$ bound in terms of derivatives of Stein’s equation. But as observed in [11], Gaussian smoothing can already provide us enough regularity to do Taylor expansions. We follow [11] in this respect.

3.1 Wasserstein distance bounds via Lindeberg’s argument

For any $n$ -th order differentiable function $f$ and $\gamma\in(0,1]$ , denote its $(n,\gamma)$ -Hölder constant by

[f]_{n,\gamma}:=\sup_{x,y:x\neq y}\frac{f^{(n)}(x)-f^{(n)}(y)}{|x-y|^{\gamma}}.

We say that $f\in C^{n,\gamma}$ if $[f]_{n,\gamma}<\infty$ .

Definition 11.

Suppose $r>1$ , and $\mu$ , $\nu$ are probability measures on $\mathbb{R}$ . Let $\ell\in\mathbb{N}$ and $\gamma\in(0,1]$ be the unique numbers such that $r=\ell+\gamma$ . We let $\Lambda_{r}=\{f\in C^{\ell,\gamma}:[f]_{\ell,\gamma}\leq 1\}$ . The Zolotarev distance $\mathcal{Z}_{r}$ is defined by

\mathcal{Z}_{r}(\mu,\nu)=\sup\left\{\int_{\mathbb{R}}f\mathrm{d}\mu-\int_{\mathbb{R}}f\mathrm{d}\nu:f\in\Lambda_{r}\right\}.

Recall that by the Kantorovich-Rubinstein Theorem, $\mathcal{W}_{r}(\mu,\nu)=\mathcal{Z}_{r}(\mu,\nu)$ for $r=1$ . When $r>1$ , it is shown by Rio [38, Theorem 3.1] that

\mathcal{W}_{r}(\mu,\nu)\lesssim_{r}\mathcal{Z}_{r}(\mu,\nu)^{1/r}.

(25)

For $\sigma>0$ and any function $f\in\Lambda_{r}$ , let

f_{\sigma}(x):=E[f(x+\sigma\mathcal{N})],

(26)

where $\mathcal{N}$ denotes a standard normal random variable. Direct integration by parts would yield the following regularity estimate for the Gaussian smoothing $f_{\sigma}$ which is a special case of [10, Lemma 6.1].

Lemma 12.

[10, Lemma 6.1] For $\sigma>0$ and $f\in\Lambda_{r}$ , $r\in\mathbb{N}$ , let $f_{\sigma}$ be as in (26). Then

\lVert f_{\sigma}^{(k)}\rVert_{\infty}\leq C_{k+1-r}\sigma^{r-k},\quad\forall k\geq r,

where $C_{n}=\int_{\mathbb{R}}\lvert u\phi^{(n)}(u)\rvert\mathrm{d}u$ , and $\phi(x)=\tfrac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ is the standard normal density.

For the reader’s convenience, we include a proof of Lemma 12 in the Appendix.

Recall the notations $\sigma_{m}(Y),X_{m}$ in (1). For any random variable $U$ , we write the expectation conditioning on $\mathcal{F}_{i-1}$ as $E_{i}$ , i.e.,

E_{i}[U]:=E[U|\mathcal{F}_{i-1}].

(27)

Proposition 13.

Let $Y=\{Y_{m}\}_{m=1}^{n}$ , $n\geq 2$ , be a martingale difference sequence and let $\beta>0$ . Assume that $\sum_{i=1}^{n}\sigma_{i}^{2}(Y)=s_{n}^{2}$ almost surely. For any $\beta>0$ , set

\lambda_{m}^{2}=\lambda_{m}^{2}(Y,\beta):=\sum_{i=m+1}^{n}\sigma_{i}^{2}(Y)+\beta^{2}s_{n}^{2}.

Then, for any $x^{3}\succcurlyeq\psi\in[0,\infty)^{[0,\infty)}$ such that $x\mapsto\psi(\sqrt{x})$ is convex,

	$\displaystyle\mathcal{Z}_{r}(X_{n},\mathcal{N})$	$\displaystyle\lesssim\beta^{r}+\frac{1}{s_{n}^{r}}E\sum_{i=1}^{n}\min\left\{\frac{E_{i}[\|Y_{i}\|^{3}]}{\lambda_{i}^{3-r}},\frac{\lambda_{i}^{r}E_{i}\psi(\|Y_{i}\|)}{\psi(\lambda_{i})}\right\},\,r\in\{1,2\},$		(28)
	$\displaystyle\mathcal{Z}_{3}(X_{n},\mathcal{N})$	$\displaystyle\lesssim\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[\|Y_{i}\|^{3}].$		(29)

Proof of Proposition 13:.

Without loss of generality , assume $s_{n}=1$ .

Let $\xi,\mathcal{N}$ be random variables with distributions $\mathcal{N}(0,\beta^{2})$ and $\mathcal{N}(0,1)$ , so that the triple $(\xi,\mathcal{N},\{X_{i}\}_{i=1}^{n})$ are independent. For $h\in\Lambda_{r}$ , $r\in\{1,2\}$ , and any random variable $U$ which is independent of $\xi$ ,

\displaystyle\lvert E[h(U+\xi)-h(U)]\rvert=\Bigr{\lvert}E[h(U+\xi)-\sum_{i=0}^{r-1}h^{(i)}(U)\xi^{i}]\Bigl{\rvert}\leq\tfrac{1}{r!}E[|\xi|^{r}].

The triangle inequality and (25) then yields, for $r\in\{1,2\}$ ,

\lvert\mathcal{Z}_{r}(X_{n},\mathcal{N})-\mathcal{Z}_{r}(X_{n}+\xi,\mathcal{N}+\xi)\rvert\lesssim\beta^{r}.

(30)

In what follows we will use Lindeberg’s argument to derive bounds for $\mathcal{Z}_{r}(X_{n}+\xi,\mathcal{N}+\xi)$ , $r\in\{1,2,3\}$ . Recall notations in (1), (27).

Note that $\lambda_{m}$ is $\mathcal{F}_{m-1}$ -measurable. Recall the function $h_{\sigma}$ in (26). Recall the operation $E_{i}$ in (27). For $h\in\Lambda_{r}$ , we consider the following telescopic sum:

\displaystyle E[h(X_{n}+\xi)-h(\mathcal{N}+\xi)]

\displaystyle=\sum_{i=1}^{n}E[h(X_{i}+\lambda_{i}\mathcal{N})-h(X_{i-1}+\lambda_{i-1}\mathcal{N})]=E\sum_{i=1}^{n}D_{i},

where $D_{i}:=E_{i}[h_{\lambda_{i}}(X_{i})-h_{\lambda_{i}}(X_{i-1}+\sigma_{i}\mathcal{N})]$ .

Note that

D_{i}=E_{i}\int_{\sigma_{i}\mathcal{N}}^{Y_{i}}\int_{0}^{t}h^{\prime\prime}_{\lambda_{i}}(X_{i-1}+s)-h^{\prime\prime}_{\lambda_{i}}(X_{i-1})\mathrm{d}s\mathrm{d}t.

(31)

Hence, for $h\in\Lambda_{r}$ , $r\in\{1,2,3\}$ , using the fact $\sigma_{i}^{3}\leq E_{i}[\lvert Y_{i}\rvert^{3}]$ , we have

|D_{i}|\lesssim E_{i}[|Y_{i}|^{3}]\lVert h_{\lambda_{i}}^{(3)}\rVert_{\infty}\stackrel{{\scriptstyle Lemma~{}\ref{lem:smoothing}}}{{\lesssim}}E_{i}[|Y_{i}|^{3}]\lambda_{i}^{r-3}.

(32)

When $r=3$ , notice that $\beta$ is irrelevant . So, letting $\beta\downarrow 0$ we immediately get (29). It remains to consider $r\in\{1,2\}$ .

Using the fact that for $f\in C^{1,1}$ ,

\displaystyle\Bigr{\lvert}\int_{0}^{y}\int_{0}^{t}f^{\prime\prime}(x+s)-f^{\prime\prime}(x)\mathrm{d}s\mathrm{d}t\Bigl{\rvert}\leq\min\left\{\tfrac{1}{6}|y|^{3}\lVert f^{(3)}\rVert_{\infty},\,y^{2}\lVert f^{\prime\prime}\rVert_{\infty}\right\},

we get, for any event $A_{i}$ and any $h\in\Lambda_{r}$ , $r\in\{1,2\}$ ,

	$\displaystyle\mathrm{I}:$	$\displaystyle=\Bigr{\lvert}E_{i}\int_{0}^{Y_{i}}\int_{0}^{t}h^{\prime\prime}_{\lambda_{i}}(X_{i-1}+s)-h^{\prime\prime}_{\lambda_{i}}(X_{i-1})\mathrm{d}s\mathrm{d}t\Bigl{\rvert}$
		$\displaystyle\leq E_{i}\left[\tfrac{1}{6}\|Y_{i}\|^{3}\lVert h_{\lambda_{i}}^{(3)}\rVert_{\infty}\mathbbm{1}_{A_{i}^{c}}+Y_{i}^{2}\lVert h_{\lambda_{i}}^{\prime\prime}\rVert_{\infty}\mathbbm{1}_{A_{i}}\right]$
		$\displaystyle\stackrel{{\scriptstyle Lemma~{}\ref{lem:smoothing}}}{{\lesssim}}E_{i}\left[\|Y_{i}\|^{3}\lambda_{i}^{r-3}\mathbbm{1}_{A_{i}^{c}}+Y_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{A_{i}}\right],$

and similarly

	$\displaystyle\mathrm{II}:$	$\displaystyle=\Bigr{\lvert}E_{i}\int_{0}^{\sigma_{i}\mathcal{N}}\int_{0}^{t}h^{\prime\prime}_{\lambda_{i}}(X_{i-1}+s)-h^{\prime\prime}_{\lambda_{i}}(X_{i-1})\mathrm{d}s\mathrm{d}t\Bigl{\rvert}$
		$\displaystyle\lesssim E_{i}\left[\sigma^{3}\lambda_{i}^{r-3}\mathbbm{1}_{A_{i}^{c}}+\sigma_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{A_{i}}\right].$

Further, since $\psi\preccurlyeq x^{3}$ , we have $s^{3}/t^{3}\leq\psi(s)/\psi(t)$ for $0<s\leq t$ . Hence

|Y_{i}|^{3}\lambda_{i}^{r-3}\mathbbm{1}_{|Y_{i}|<\lambda_{i}}\leq\lambda_{i}^{r}\frac{\psi(|Y_{i}|)}{\psi(\lambda_{i})}\mathbbm{1}_{|Y_{i}|<\lambda_{i}}.

Since $\psi\succcurlyeq x^{2}$ , we have $s^{2}/t^{2}\leq\psi_{1}(s)/\psi_{1}(t)$ for $0<t\leq s$ . Thus

Y_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{|Y_{i}|\geq\lambda_{i}}\leq\lambda_{i}^{r}\frac{\psi(|Y_{i}|)}{\psi(\lambda_{i})}\mathbbm{1}_{|Y_{i}|\geq\lambda_{i}},

and so, for $r\in\{1,2\}$ ,

\displaystyle\mathrm{I}\lesssim E_{i}\left[|Y_{i}|^{3}\lambda_{i}^{r-3}\mathbbm{1}_{|Y_{i}|<\lambda_{i}}+Y_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{|Y_{i}|\geq\lambda_{i}}\right]\lesssim\frac{\lambda_{i}^{r}E_{i}[\psi(|Y_{i}|)]}{\psi(\lambda_{i})}.

Similarly, for $h\in\Lambda_{r}$ , $r\in\{1,2\}$ ,

\mathrm{II}\lesssim E_{i}\left[\sigma^{3}\lambda_{i}^{r-3}\mathbbm{1}_{\sigma_{i}<\lambda_{i}}+\sigma_{i}^{2}\lambda_{i}^{r-2}\mathbbm{1}_{\sigma_{i}\geq\lambda_{i}}\right]\lesssim\frac{\lambda_{i}^{r}\psi(\sigma_{i})}{\psi(\lambda_{i})}.

Since $x\mapsto\psi(\sqrt{x})$ is convex, by Jensen’s inequality we know that

\psi(\sigma_{i})\leq E_{i}[\psi(|Y_{i}|)].

(33)

Thus, we obtain, for $h\in\Lambda_{r}$ , $r\in\{1,2\}$ ,

|D_{i}|\stackrel{{\scriptstyle\eqref{eq:di-bound}}}{{\leq}}\mathrm{I}+\mathrm{II}\lesssim\frac{\lambda_{i}^{r}E_{i}[\psi(|Y_{i}|)]}{\psi(|\lambda_{i}|)}.

This inequality, together with (32), (30) and (25), yields the Proposition. ∎

Remark 14.

If there exists $1\leq j\leq n$ such that, given $\mathcal{F}_{j-1}$ , the distribution of $Y_{j}$ is the normal $\mathcal{N}(0,\sigma_{j}^{2})$ , then the $j$ -th summand in (28) and (29) can be removed. That is, the summations in (28) and (29) can both be replaced by

\sum_{i=1,i\neq j}^{n}.

Indeed, in this case, $X_{j}$ and $X_{j-1}+\sigma_{j}\mathcal{N}$ are identically distributed conditioning on $\mathcal{F}_{j-1}$ . Thus $D_{j}=0$ .

3.2 The rates of some special cases

From Proposition 13 we can obtain Wasserstein distance bounds for some special cases (with $V_{n}^{2}=1$ ). See Corollary 15 below. These cases usually yield better rates than the typical cases. They are not only of interest on their own right, but can also serve as important references when we construct counterexamples.

The $\mathcal{W}_{1}$ bounds within Corollary 15 can be considered generalizations of some of the results in [39, Corollaries 2.2,2.3] and [19, Corollaries 2.5,2.6,2.7] to the $\mathscr{L}^{\psi}$ integrable cases.

Corollary 15.

Assume that $V_{n}^{2}=1$ almost surely. Let $\psi$ be an N-function such that $\psi\preccurlyeq x^{3}$ and $x\mapsto\psi(\sqrt{x})$ is convex. Then the following statements hold.

(i)

$\mathcal{W}_{r}(X_{n},\mathcal{N})\lesssim L_{\psi}$ for $r\in\{1,2\}$ .
(ii)
If there exists $\sigma>0$ such that

$P(\sigma_{i}\geq\sigma\,\text{ for all }1\leq i\leq n)=1,$

then, writing $M_{\psi}=\max_{i=1}^{n}E[\psi(|Y_{i}|)]$ (and write $M_{\psi}$ as $M_{3}$ when $\psi=x^{3}$ ),
- •
  
  $\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\left\{\begin{array}[]{lr}\frac{M_{3}}{\sigma^{2}s_{n}}\log n&\text{ when }\psi=x^{3},\\ (3-p)^{-1}\frac{M_{\psi}s_{n}^{2}}{\sigma^{2}\psi(s_{n})}&\text{ when }\psi\preccurlyeq x^{p},p\in(2,3);\end{array}\right.$ .
- •
  
  $\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\frac{M_{\psi}^{1/2}s_{n}}{\sigma\psi(s_{n})^{1/2}}$ .

(iii)

If there exists a constant $\theta>0$ such that almost surely,

E[\psi(|Y_{i}|)|\mathcal{F}_{i-1}]\leq\theta\sigma_{i}^{2}\quad\forall i=1,\ldots,n,

then

•

$\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim_{\theta}\left\{\begin{array}[]{lr}\frac{1}{s_{n}}\log(e+s_{n})&\text{ when }\psi=x^{3},\\ \frac{s_{n}^{2}}{\psi(s_{n})}&\text{ when }\psi\preccurlyeq x^{p},p\in(2,3);\end{array}\right.$
•

$\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim_{\theta}\frac{s_{n}}{\psi(s_{n})^{1/2}}$ .

In particular, when $\theta=\max_{i=1}^{n}\lVert Y_{i}\rVert_{\infty}<\infty$ , we have

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\frac{\theta}{s_{n}}\log(e+s_{n})\,\text{ and }\,\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim(\tfrac{\theta}{s_{n}})^{1/2}.

Proof.

(i) For the ease of notation we write $\theta:=\lVert Y\rVert_{\psi}$ . Note that $x\mapsto\psi(x/\theta)$ is still an N-function that satisfies conditions of Proposition 13. Recall the conditional expectation $E_{i}$ in (27). By Proposition 13,

	$\displaystyle\mathcal{Z}_{r}(X_{n},\mathcal{N})$	$\displaystyle\lesssim\beta^{r}+\frac{1}{s_{n}^{r}}E\sum_{i=1}^{n}\frac{\lambda_{i}^{r}E_{i}[\psi(\|Y_{i}\|/\theta)]}{\psi(\lambda_{i}/\theta)}$
		$\displaystyle\lesssim\beta^{r}+\frac{1}{s_{n}^{r}}E\sum_{i=1}^{n}\frac{(\beta s_{n})^{r}E_{i}[\psi(\|Y_{i}\|/\theta)]}{\psi(\beta s_{n}/\theta)}$
		$\displaystyle\lesssim\beta^{r}+\frac{n\beta^{r}}{\psi(\beta s_{n}/\theta)}\quad\text{ for any }\beta>0,r\in\{1,2\},$

where in the second inequality we used the fact that $\lambda_{i}\geq\beta s_{n}$ for all $i$ and that $x\mapsto\frac{x^{r}}{\psi(x)}$ is decreasing. Taking $\beta=\frac{\theta}{s_{n}}\psi^{-1}(n)$ , the $\mathcal{Z}_{1}$ , $\mathcal{Z}_{2}$ bounds are proved.

(ii) Recall $\lambda_{i}$ in Proposition 13. When $\sigma_{i}\geq\sigma$ for all $i$ we have

\lambda_{i}^{2}\geq(n-i)\sigma^{2}+\beta^{2}s_{n}^{2},\quad\forall 1\leq i\leq n.

Thus, by Proposition 13, we get, for any $\beta\geq 2\sigma/s_{n}$ , $x^{r}\preccurlyeq\psi\preccurlyeq x^{3}$ , $r\in\{1,2\}$ ,

	$\displaystyle\mathcal{Z}_{r}(X_{n},\mathcal{N})$	$\displaystyle\lesssim\beta^{r}+\frac{M_{\psi}}{s_{n}^{r}}\sum_{i=1}^{n}\tfrac{[(n-i)\sigma^{2}+\beta^{2}s_{n}^{2}]^{r/2}}{\psi(\sqrt{(n-i)\sigma^{2}+\beta^{2}s_{n}^{2}})}$
		$\displaystyle\lesssim\beta^{r}+\frac{M_{\psi}}{s_{n}^{r}\sigma^{2}}\int_{\sigma}^{\sqrt{n}\sigma+\beta s_{n}}\frac{t^{r+1}}{\psi(t)}\mathrm{d}t.$		(34)

The first $\mathcal{Z}_{1}$ bound follows when $r=1$ , $\psi(x)=x^{3}$ , $\beta=2M_{3}/(s_{n}\sigma^{2})$ .

Now consider the case $x^{r}\preccurlyeq\psi\preccurlyeq x^{p}$ , $p\in(2,r+2)\cap(2,3]$ , $r\in\{1,2\}$ . Since $s_{n}^{2}\geq n\sigma$ and $x\mapsto x^{p}/\psi(x)$ is increasing, we get, for any $\beta\in[\tfrac{2\sigma}{s_{n}},1]$ ,

	$\displaystyle\int_{\sigma}^{\sqrt{n}\sigma+\beta s_{n}}\frac{t^{p}}{\psi(t)}t^{r+1-p}\mathrm{d}t$	$\displaystyle\leq\frac{(2s_{n})^{p}}{\psi(2s_{n})}\int_{0}^{2s_{n}}t^{r+1-p}\mathrm{d}t$
		$\displaystyle\leq\frac{1}{r+2-p}\frac{(2s_{n})^{r+2}}{\psi(2s_{n})}$
		$\displaystyle\leq\frac{1}{r+2-p}\frac{2^{r}s_{n}^{r+2}}{\psi(s_{n})}.$

Taking $\beta$ such that $\beta^{r}=C\frac{M_{\psi}}{\sigma^{2}}\frac{s_{n}^{2}}{\psi(s_{n})}$ for appropriate constant $C>0$ and recalling (3.2), we get (the case $x^{r}\preccurlyeq\psi\preccurlyeq x^{p}$ , $p\in(2,r+2)\cap(2,3]$ )

\mathcal{Z}_{r}(X_{n},\mathcal{N})\lesssim\frac{1}{r+2-p}\beta^{r}.

the rest of the $\mathcal{W}_{r}$ bounds in (ii) follow. Note that this is a trivial inequality (since $\mathcal{W}_{r}(X_{n},\mathcal{N})\lesssim 1$ ) if $\beta>1$ . If $\beta<1$ , it remains to justify that such a choice of $\beta$ satisfies $\beta\geq 2\sigma/s_{n}$ . Indeed, since $\psi\succcurlyeq x^{2}$ and $\sigma_{i}\geq\sigma$ , we have, for $1\leq i\leq n$ ,

M_{\psi}\geq\frac{\psi(\sigma/2)}{(\sigma/2)^{2}}E[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|\geq\sigma/2}]\gtrsim\psi(\sigma/2)

and so, for $r\in\{1,2\}$ ,

\beta s_{n}\geq\beta^{r}s_{n}\gtrsim\frac{\psi(\sigma/2)s_{n}^{3}}{(\sigma/2)^{2}\psi(s_{n})}\gtrsim\sigma,

where in the last inequality we used the fact that $\psi\lesssim x^{3}$ .

(iii) First, notice that there exists $\gamma>0$ such that $\sigma_{i}<\gamma$ , $\forall 1\leq i\leq n$ . Indeed,

\displaystyle\frac{3}{4}\sigma_{i}^{2}

\displaystyle\leq E_{i}[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|>\sigma_{i}/2}]\leq E_{i}[\psi(|Y_{i}|)]/g(\tfrac{\sigma_{i}}{2})\leq\theta\sigma_{i}^{2}/g(\tfrac{\sigma_{i}}{2})

where $g(x):=\psi(x)/x^{2}$ is an increasing function on $(0,\infty)$ . Thus

\sigma_{i}\leq 2g^{-1}(\tfrac{4\theta}{3})=:\gamma.

Next, by Proposition 13 and $E_{i}[\psi(|Y_{i}|)]\leq\theta\sigma_{i}^{2}$ , we get

\mathcal{Z}_{r}(X_{n},\mathcal{N})\lesssim\beta^{r}+\frac{\theta}{s_{n}^{r}}E\sum_{i=1}^{n}\frac{\lambda_{i}^{r}\sigma_{i}^{2}}{\psi(\lambda_{i})}.

Define a sequence of stopping times $\{\tau_{t}\}_{0\leq t\leq 1}$ by

\tau(t)=\tau_{t}:=\inf\{m\geq 1:\sum_{i=1}^{m}\sigma_{i}^{2}\geq t\}.

(35)

Clearly, $\tau_{t}=m$ if and only if $t\in(\sum_{i=1}^{m-1}\sigma_{i}^{2},\sum_{i=1}^{m}\sigma_{i}^{2}]$ . Moreover,

\lambda_{\tau_{t}}^{2}=s_{n}^{2}(1+\beta^{2})-\sum_{i=1}^{\tau_{t}}\sigma_{i}^{2}\geq s_{n}^{2}(1+\beta^{2})-t-\sigma_{\tau_{t}}^{2}\geq s_{n}^{2}(1+\beta^{2})-t-\gamma^{2}.

Thus we get, for any $\beta\in[\frac{\gamma}{s_{n}},1]$ , $r\in\{1,2\}$ ,

\displaystyle\sum_{i=1}^{n}\frac{\lambda_{i}^{r}\sigma_{i}^{2}}{\psi(\lambda_{i})}=\int_{0}^{s_{n}^{2}}\frac{\lambda_{\tau_{t}}^{r}}{\psi(\lambda_{\tau_{t}})}\mathrm{d}t\leq\int_{s_{n}^{2}\beta^{2}-\gamma^{2}}^{(1+\beta^{2})s_{n}^{2}}\frac{t^{r/2}}{\psi(t^{1/2})}\mathrm{d}t\leq 2\int_{\sqrt{s_{n}^{2}\beta^{2}-\gamma^{2}}}^{\sqrt{2}s_{n}}\frac{t^{r+1}}{\psi(t)}\mathrm{d}t,

where in the first inequality we used the fact that $x\mapsto x/\psi(x)$ is decreasing. When $\psi=x^{3}$ , taking $\beta=2\gamma/s_{n}$ , the desired $\mathcal{W}_{1}$ bound follows. For the case $\psi\preccurlyeq x^{p}$ , $2<p<3$ , the bound of this integral can be handled the same way as in (ii). Then the $\mathcal{W}_{1}$ bounds in (iii) are proved.

It remains to prove the $\mathcal{Z}_{2}$ bound. By the inequality above, for any $\beta\in[\frac{\gamma}{s_{n}},1]$ ,

\displaystyle\mathcal{Z}_{2}(X_{n},\mathcal{N})\lesssim\beta^{2}+\frac{\theta}{s_{n}^{2}}\int_{0}^{\sqrt{2}s_{n}}\frac{t^{3}}{\psi(t)}\mathrm{d}t\lesssim\beta^{2}+\frac{\theta s_{n}^{2}}{\psi(s_{n})},

where in the last inequality we used the fact that $x\mapsto x^{3}/\psi$ is increasing. Taking $\beta^{2}=\frac{s_{n}^{2}\psi(\gamma)}{\psi(s_{n})\gamma^{2}}$ , the $\mathcal{Z}_{2}$ bound in (iii) follows. ∎

4 Proofs of the main $\mathcal{W}_{r}$ bounds

This section is devoted to the proofs of Theorems 5, 7, and 8.

Note that the Taylor expansion results (Proposition 13), which are in terms of the second and third conditional moments, suit exactly martingales (with $V_{n}^{2}=1$ ) with integrability between $\mathscr{L}^{2}$ and $\mathscr{L}^{3}$ , and so not surprisingly, the $\mathcal{W}_{1},\mathcal{W}_{2}$ bounds of Corollary 15 for $\mathscr{L}^{p}$ martingales, $p\in(2,3]\cup\{\infty\}$ , followed quite easily from Proposition 13. However, to obtain Wasserstein bounds for martingales with more general integrability, we need new insights.

In Subsection 4.1, we will modify the mds into a bounded sequence with deterministic total conditional variance $V_{n}^{2}$ . To this end, we first truncate the martingale into a bounded sequence with $V_{n}^{2}\leq 1$ , and then lengthen it to have $V_{n}^{2}=1$ . Such tricks of elongation and truncation of martingales were already used by Bolthausen [5] (the idea goes back to Dvoretzky [13]) and Haeusler [25] in the study of the Kolmogorov distance of the martingale CLT. The main result of Subsection 4.1 is a control of the error due to this modification in terms of $L_{\psi}$ and $(V_{n}^{2}-1)$ . See Proposition 17. As an application, we prove the $\mathcal{W}_{3}$ bound in Theorem 8.

In Subsections 4.2 and 4.3, we prove the $\mathcal{W}_{1},\mathcal{W}_{2}$ estimates in Theorems 5 and 7 by bounding the corresponding metrics for the modified martingale. A crucial step of our method is to use Young’s inequality inside the conditional expectations to “decouple” the Lyapunov coefficient $L_{\psi}$ and the conditional variances $\sigma_{i}^{2}$ from the summation within Proposition 13. This will turn the $\mathcal{W}_{r}$ bounds into an optimization problem over three parameters: the truncation parameter ( $\alpha$ ), the smoothing parameter ( $\beta$ ), and an Orlicz parameter ( $\rho$ ). To this end, tools from the theory of N-functions will be employed to compare N-functions to polynomials.

4.1 A modified martingale, and Proof of Theorem 8 ( $\mathcal{W}_{3}$ bounds)

The goal of this subsection is to modify the original mds $\{Y_{i}\}_{i=1}^{n}$ to a new mds $\hat{Y}=\{\hat{Y}_{i}\}_{i=1}^{n+1}$ which is uniformly bounded except the last term. Throughout this subsection, we let $\alpha\in(0,\infty]$ be any fixed constant.

Define a martingale difference sequence $Z=Z^{(\alpha)}=\{Z_{i}\}_{i=1}^{n}$ as

Z_{i}=Z_{i}^{(\alpha)}:=Y_{i}\mathbbm{1}_{|Y_{i}|\leq\alpha/2}-E[Y_{i}\mathbbm{1}_{|Y_{i}|\leq\alpha/2}|\mathcal{F}_{i-1}],\quad 1\leq i\leq n.

(36)

Note that $P(|Z_{i}|\leq\alpha)=1$ for all $1\leq i\leq n$ and, with $\sigma_{i}^{2}(Z):=E[Z_{i}^{2}|\mathcal{F}_{i-1}]$ ,

\displaystyle\sum_{i=1}^{n}\sigma_{i}^{2}(Z)

\displaystyle\leq\sum_{i=1}^{n}E[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|\leq\alpha/2}]\leq\sum_{i=1}^{n}\sigma_{i}^{2}(Y).

(37)

Define a stopping time (with the convention $\inf\emptyset=\infty$ )

T=\inf\big{\{}1\leq m\leq n:\sum_{i=1}^{m}\sigma_{i}^{2}(Z)>s_{n}^{2}\big{\}}\wedge(n+1).

(38)

Since $\sum_{i=1}^{m}\sigma_{i}^{2}(Z)$ is $\mathcal{F}_{m-1}$ -measurable, $(T-1)$ is also a stopping time.

Definition 16.

Let $\alpha\in(0,\infty]$ and let $\xi\sim\mathcal{N}(0,1)$ be a standard normal which is independent of $\{Y_{i}\}_{i=1}^{n}$ . Define the modified martingale difference sequence $\hat{Y}=\hat{Y}^{(\alpha)}=\{\hat{Y}_{i}\}_{i=1}^{\infty}$ as follows: $\forall i\geq 1$ ,

\hat{Y}_{i}=\hat{Y}_{i}^{(\alpha)}:=Z_{i}\mathbbm{1}_{\sum_{j=1}^{i}\sigma_{j}^{2}(Z)\leq 1}+\xi\left(s_{n}^{2}-\sum_{j=1}^{T-1}\sigma_{j}^{2}(Z)\right)^{1/2}\mathbbm{1}_{i=n+1}.

(39)

Enlarging the $\sigma$ -fields $\{\mathcal{F}_{i}\}$ if necessary, $\{\hat{Y}_{m}\}_{m\geq 0}$ is still a mds. Let

\hat{X}_{m}=\frac{1}{s_{n}}\sum_{i=1}^{m}\hat{Y}_{i},\quad m\geq 1,\text{ with }\hat{X}_{0}=0.

Note that $\hat{Y}_{j}=0$ for all $j\in[T,n]\cup(n+1,\infty)$ and recall that $T\leq n+1$ . We write, for $m\geq 1$ , $\sigma_{m}^{2}(\hat{Y}):=E[\hat{Y}_{m}^{2}|\mathcal{F}_{m-1}]$ .

The main feature of the mds $\hat{Y}$ is that $|\hat{Y}_{i}|\leq\alpha$ a.s. for $1\leq i\leq n$ , and

\sum_{i=1}^{n+1}\sigma_{i}^{2}(\hat{Y})=s_{n}^{2}\quad\text{ almost surely.}

(40)

Proposition 17.

For $\alpha\in(0,\infty]$ and a mds $\{Y_{i}\}_{i=1}^{n}$ , let $\{\hat{Y}_{i}\}$ and $Z=\{Z_{i}\}_{i=1}^{n}$ be as in Definition 16. Let $r\geq 1$ and let $\psi\in[0,\infty)^{[0,\infty)}$ be an N-function with $\psi\succcurlyeq x^{r\vee 2}$ . Recall the definition of $L_{\psi}=L_{\psi}(Y)$ in (20). Set $\theta_{\psi}:=\lVert Y\rVert_{\psi}$ .

(a)

For $\alpha<\infty$ , $r\geq 1$ ,

\displaystyle\lvert\mathcal{W}_{r}(X_{n},\mathcal{N})-\mathcal{W}_{r}(\hat{X}_{n+1},\mathcal{N})\rvert\lesssim_{r}L_{\psi}+\alpha+\frac{n^{1/2}\alpha}{\psi(\alpha s_{n}/2\theta_{\psi})^{1/(r\vee 2)}}+\lVert V_{n}^{2}-1\rVert_{r/2}^{1/2}.

(b)

If $\psi$ satisfies that $x\mapsto\psi(\sqrt{x})$ is convex, then for $r\geq 1$ ,

\displaystyle\lvert\mathcal{W}_{r}(X_{n},\mathcal{N})-\mathcal{W}_{r}(\hat{X}_{n+1},\mathcal{N})\rvert\lesssim_{r}L_{\psi}+\frac{n^{1/2}\alpha}{\psi(\alpha s_{n}/2\theta_{\psi})^{1/(r\vee 2)}}\mathbbm{1}_{\alpha<\infty}+\lVert V_{n}^{2}-1\rVert_{r/2}^{1/2}.

Estimate (b) is slightly better than (a) but with a slightly stronger condition. Statement (b) is more useful for the case $\alpha=\infty$ .

Lemma 18.

Let $Z=\{Z_{i}\}_{i=1}^{n}$ be as in (36). Recall the definition of the orlicz norm $\lVert\cdot\rVert_{\psi}$ for a sequence in Definition 3. Then, for any N-function $\psi$ ,

\lVert Z\rVert_{\psi}\leq 2\lVert Y\rVert_{\psi}

Proof.

Without loss of generality, assume $\lVert Y\rVert_{\psi}=1$ . It suffices to show that

E\sum_{i=1}^{n}\psi(|Z_{i}|/2)\leq n.

(41)

Indeed, recall $E_{i}[\cdot]$ in (27). By the definition of $Z_{i}$ in (36), for $1\leq i\leq n$ ,

	$\displaystyle E_{i}[\psi(\|Z_{i}\|/2)]$	$\displaystyle\leq E_{i}\left[\psi\left((\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2}+E_{i}[\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2}])/2\right)\right]$
		$\displaystyle\leq\frac{1}{2}E_{i}\left[\psi(\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2})+\psi(E_{i}[\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2}])\right]$
		$\displaystyle\leq E_{i}\left[\psi(\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2})\right],$

where we used Jensen’s inequality in both the second and the third inequalities. Summing both sides over all $1\leq i\leq n$ , inequality (41) follows. ∎

Proof of Proposition 17.

Throughout, notice that $r\geq 1$ and $\psi\succcurlyeq x^{r\vee 2}$ .

Without loss of generality, assume $s_{n}=1$ . Set

\displaystyle K_{r}:=\lVert V_{n}^{2}-1\rVert_{r/2}^{1/2}.

(42)

To prove Proposition 17, it suffices to show that

\lVert X_{n}-\hat{X}_{n+1}\rVert_{r}\lesssim_{r}L_{\psi}+\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}+K_{r}+\frac{n^{1/2}\alpha}{\psi(\alpha/2\theta_{\psi})^{1/(r\vee 2)}}\mathbbm{1}_{\alpha<\infty}.

(43)

Indeed, trivially we have $\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}\leq\alpha$ . If $\psi(\sqrt{x})$ is convex, we claim that

\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}\lesssim L_{\psi}.

(44)

To this end, we apply Lemma A.2 to get

\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}\leq 2^{1/r}\lVert\{\sigma_{i}(Z)\}_{i=1}^{n}\rVert_{\psi}\psi^{-1}(n)

Notice that, when $\psi(\sqrt{x})$ is convex, then using Jensen’s inequality as in (33), we have $\lVert\{\sigma_{i}(Z)\}_{i=1}^{n}\rVert_{\psi}\leq\lVert Z\rVert_{\psi}$ which is bounded by $2\lVert Y\rVert_{\psi}$ by Lemma 18. Claim (44) follows.

The rest of the proof is devoted to obtain (43).

Recall $T$ in (38). Since $\{\hat{Y}_{i}\}$ and $\{Z_{i}\}$ coincide up to $i=T-1$ , we have

\displaystyle X_{n}-\hat{X}_{n+1}

\displaystyle=\sum_{i=T}^{n}Z_{i}+\sum_{i=1}^{n}(Y_{i}-Z_{i})-\hat{Y}_{n+1}.

(45)

Step 1. We will bound the first term $\sum_{i=T}^{n}Z_{i}$ in (45) by

\lVert\sum_{i=T}^{n}Z_{i}\rVert_{r}\lesssim_{r}K_{r}+L_{\psi},\quad r\geq 1.

(46)

Since $T-1$ is a stopping time, $\{Z_{i}\}_{i=T}^{n}$ form a martingale difference sequence. Hence, by an inequality of Burkholder Theorem A.3 and Lemma A.2, for $r\geq 1$ ,

\displaystyle\lVert\sum_{i=T}^{n}Z_{i}\rVert_{r}\leq\lVert\sum_{i=T+1}^{n}Z_{i}\rVert_{r}+\lVert\max_{i=1}^{n}|Z_{i}|\rVert_{r}\lesssim_{r}\lVert\sum_{i=T+1}^{n}\sigma_{i}^{2}(Z)\rVert_{r/2}^{1/2}+L_{\psi}.

(47)

Further, by the definition of the stopping time $T$ ,

\displaystyle\sum_{i=T+1}^{n}\sigma_{i}^{2}(Z)=\left(\sum_{i=1}^{n}\sigma_{i}^{2}(Z)-\sum_{i=1}^{T}\sigma_{i}^{2}(Z)\right)\vee 0\leq\left(\sum_{i=1}^{n}\sigma_{i}^{2}(Z)-1\right)\vee 0.

This inequality, together with (47), yields (46).

Step 2. Consider the second $\sum_{i=1}^{n}(Y_{i}-Z_{i})$ in (45). We will show that

\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})\rVert_{r}\lesssim_{r}\frac{\alpha n^{1/2}}{\psi(\alpha/2\theta_{\psi})^{1/(r\vee 2)}}\mathbbm{1}_{\alpha<\infty}.

(48)

Indeed, this inequality is trivial for $\alpha=\infty$ . When $\alpha<\infty$ , note that $\{Y_{i}-Z_{i}\}_{i=1}^{n}$ is a martingale difference sequence. For $r\geq 1$ , by Burkholder’s inequality,

\displaystyle\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})\rVert_{r}\lesssim_{r}\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})^{2}\rVert_{r/2}^{1/2}

Further, by Hölder’s inequality, Jensen’s inequality, and the fact $Y_{i}-Z_{i}=Y_{i}\mathbbm{1}_{|Y_{i}|>\alpha/2}-E[Y_{i}\mathbbm{1}_{|Y_{i}|>\alpha/2}|\mathcal{F}_{i-1}]$ , we get, for $r\geq 2$ ,

\displaystyle\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})^{2}\rVert_{r/2}\leq n\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}-Z_{i}|^{r}\bigg{\rVert}_{1}^{2/r}\leq 4n\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\bigg{\rVert}_{1}^{2/r}.

Since $|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\leq\psi(|Y_{i}|/\theta_{\psi})(\alpha/2)^{r}/\psi(\alpha/2\theta_{\psi})$ , we have

\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|>\alpha/2}\bigg{\rVert}_{1}\leq\frac{(\alpha/2)^{r}}{n\psi(\alpha/2\theta_{\psi})}\sum_{i=1}^{n}E[\psi(|Y_{i}|/2\theta_{\psi})]\leq\frac{(\alpha/2)^{r}}{\psi(\alpha/2\theta_{\psi})}

(49)

where the last inequality used the definition of $\lVert Y\rVert_{\psi}$ . Hence

\lVert\sum_{i=1}^{n}(Y_{i}-Z_{i})^{2}\rVert_{r/2}\leq\frac{\alpha^{2}n}{\psi(\alpha/2\theta_{\psi})^{2/(r\vee 2)}}.

Inequality (48) is proved.

Step 3. Consider the last term in (45). By definition (39),

\lVert\hat{Y}_{n+1}\rVert_{r}\lesssim_{r}\bigg{\lVert}\big{(}1-\sum_{i=1}^{T-1}\sigma_{i}^{2}(Z)\big{)}^{1/2}\bigg{\rVert}_{r}.

When $T\leq n$ , we have

0\leq 1-\sum_{i=1}^{T-1}\sigma_{i}^{2}(Z)\leq 1-\sum_{i=1}^{T}\sigma_{i}^{2}(Z)+\max_{i=1}^{n}\sigma_{i}^{2}(Z)<\max_{i=1}^{n}\sigma_{i}^{2}(Z).

If $T=n+1$ , then

\Bigr{\lvert}1-\sum_{i=1}^{T-1}\sigma_{i}^{2}(Z)\Bigl{\rvert}\leq\Bigr{\lvert}1-\sum_{i=1}^{n}\sigma_{i}^{2}(Y)\Bigl{\rvert}+\Bigr{\lvert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sum_{i=1}^{n}\sigma_{i}^{2}(Z)\Bigl{\rvert}.

Hence, we have

\lVert\hat{Y}_{n+1}\rVert_{r}\leq\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}+K_{r}+\bigg{\lVert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)\bigg{\rVert}_{r/2}^{1/2}.

(50)

Step 4. Let us bound the third term in (50). We will show that

\bigg{\lVert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)\bigg{\rVert}_{r/2}\lesssim\frac{n\alpha^{2}}{\psi(\alpha/2\theta_{\psi})^{2/(r\vee 2)}}\mathbbm{1}_{\alpha<\infty}.

(51)

This is trivial when $\alpha=\infty$ . For $\alpha<\infty$ , recall the notation $E_{i}$ in (27). Then

\displaystyle\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)=E_{i}[Y_{i}^{2}\mathbbm{1}_{|Y_{i}|>\alpha/2}]+E_{i}[Y_{i}\mathbbm{1}_{|Y_{i}|>\alpha/2}]^{2}\quad\forall 1\leq i\leq n.

Hence, by Hölder’s inequality and Jensen’s inequality, for $r\geq 2$ ,

	$\displaystyle\bigg{\lVert}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)-\sigma_{i}^{2}(Z)\bigg{\rVert}_{r/2}$	$\displaystyle\leq 2\bigg{\lVert}\sum_{i=1}^{n}E_{i}[Y_{i}^{2}\mathbbm{1}_{\|Y_{i}\|>\alpha/2}]\bigg{\rVert}_{r/2}$
		$\displaystyle\leq 2n\bigg{\lVert}\frac{1}{n}\sum_{i=1}^{n}\|Y_{i}\|^{r}\mathbbm{1}_{\|Y_{i}\|>\alpha/2}\bigg{\rVert}_{1}^{2/r}\stackrel{{\scriptstyle\eqref{eq:noname}}}{{\lesssim}}\frac{n\alpha^{2}}{\psi(\alpha/2\theta_{\psi})^{2/r}}.$

Display (51) is proved.

Step 5. By (50), (51), we have arrived at

\lVert\hat{Y}_{n+1}\rVert_{r}\lesssim\lVert\max_{i}^{n}\sigma_{i}(Z)\rVert_{r}+K_{r}+\frac{\alpha n^{1/2}}{\psi(\alpha/2\theta_{\psi})^{1/(r\vee 2)}}\mathbbm{1}_{\alpha<\infty}.

This inequality, together with (45), (46), (48), yields (43). ∎

Proof of Theorem 8.

Consider $\alpha=\infty$ and recall the definition of $\hat{Y}=\hat{Y}^{(\infty)}$ in Definition 16. Note that $Z_{i}^{(\infty)}=Y_{i}$ for $1\leq i\leq n$ .

Since $\hat{Y}_{n+1}$ is normal conditioning on $\mathcal{F}_{n}$ , by (29) in Proposition 13 and Remark 14, we have

\mathcal{Z}_{3}(\hat{X}_{n+1},\mathcal{N})\lesssim\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|\hat{Y}_{i}|^{3}]\leq\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|Z_{i}|^{3}]=\frac{1}{s_{n}^{3}}\sum_{i=1}^{n}E[|Y_{i}|^{3}].

By (25), this implies $\mathcal{W}_{3}(\hat{X}_{n+1},\mathcal{N})\lesssim L_{3}$ . Further, applying Proposition 17(b) to the case $\alpha=\infty$ , we get

\mathcal{W}_{3}(X_{n},\mathcal{N})\lesssim\mathcal{W}_{3}(\hat{X}_{n+1},\mathcal{N})+L_{3}+\lVert V_{n}^{2}-1\rVert_{3/2}^{1/2}.

Theorem 8 is proved. ∎

4.2 Proof of Theorem 5 ( $\mathcal{W}_{1}$ bounds)

In what follows we let $\alpha\in(0,\infty),\rho>0$ be constants to be determined later. Let the mds $\hat{Y}=\hat{Y}^{(\alpha)}$ be as in Definition 16, and set

\beta=\sqrt{2}\alpha.

(52)

Throughout this subsection, we simply write

	$\displaystyle\sigma_{i}^{2}$	$\displaystyle=\sigma_{i}^{2}(\hat{Y})=E[\hat{Y}_{i}^{2}\|\mathcal{F}_{i-1}],\quad i\geq 1$
	$\displaystyle\lambda_{m}^{2}$	$\displaystyle=\sum_{i=m+1}^{n+1}\sigma_{i}^{2}+\beta^{2}s_{n}^{2},\quad 0\leq m\leq n+1.$		(53)

Note that $\lambda_{0}^{2}=(1+\beta^{2})s_{n}^{2}$ , and $\lambda_{m}^{2}=\lambda_{0}^{2}-\sum_{i=1}^{m}\sigma_{i}^{2}$ is $\mathcal{F}_{m-1}$ -measurable for $m\geq 1$ .

Recall that $\hat{Y}_{n+1}$ is normaly distributed given $\mathcal{F}_{i-1}$ . By (40), Proposition 13, and Remark 14, for $r\in\{1,2\}$ , $\alpha>0,\beta=\sqrt{2}\alpha$ ,

\mathcal{W}_{r}(\hat{X}_{n+1},\mathcal{N})\lesssim\alpha^{r}+s_{n}^{-r}E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}^{3-r}}.

(54)

The following estimate of the summation within (54) will be crucially employed in the derivation of our $\mathcal{W}_{r}$ bounds, $r=1,2$ .

Proposition 19.

Let $\psi\in[0,\infty)^{[0,\infty)}$ be an N-function with $\psi\succcurlyeq x^{2}$ , and set $\theta_{\psi}=\lVert Y\rVert_{\psi}$ . Recall $\alpha,\hat{Y},\lambda_{i}$ above, and $\beta=\sqrt{2}\alpha$ . Then, for any $B,k>0$ ,

E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}^{k}}\mathbbm{1}_{\lambda_{i}^{2}\leq B}\lesssim\theta_{\psi}\rho n+\theta_{\psi}\rho\alpha^{-2}\int_{\alpha^{2}}^{(1+\alpha^{2})\wedge B}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{k/2}})\mathrm{d}t,\quad\forall\rho>0.

Proof.

Without loss of generality, assume $s_{n}=1$ .

Recall that $\lambda_{i}$ is $\mathcal{F}_{i-1}$ -measurable. Recall that $E_{i}[\cdot]=E[\cdot|\mathcal{F}_{i-1}]$ as in (27).

By Young’s inequality (15), for any $\rho>0$ ,

E\sum_{i=1}^{n}\frac{E_{i}[|\hat{Y}_{i}|^{3}]}{2\theta_{\psi}\rho\lambda_{i}^{k}}\mathbbm{1}_{\lambda_{i}^{2}\leq B}\leq E\sum_{i=1}^{n}E_{i}\big{[}\psi(\tfrac{|\hat{Y}_{i}|}{2\theta_{\psi}})+\psi_{*}(\tfrac{\hat{Y}_{i}^{2}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq B}\big{]}.

Note that $\psi_{*}$ is an N-function and so $\psi_{*}\succcurlyeq x$ . Using the fact that $\psi_{*}\succcurlyeq x$ and that $|\hat{Y}_{i}|\leq\alpha$ for $1\leq i\leq n$ , we have, almost surely,

\psi_{*}(\tfrac{\hat{Y}_{i}^{2}}{\rho\lambda_{i}^{k}})\leq\tfrac{\hat{Y_{i}}^{2}}{\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i}^{k}}),\quad\forall 1\leq i\leq n.

Thus we further have (Note that $\hat{Y}_{i}=0$ for $T\leq i\leq n$ .)

	$\displaystyle E\sum_{i=1}^{n}\frac{E_{i}[\|\hat{Y}_{i}\|^{3}]}{2\theta_{\psi}\rho\lambda_{i}^{k}}\mathbbm{1}_{\lambda_{i}^{2}\leq B}$	$\displaystyle\leq E\sum_{i=1}^{n}\psi(\tfrac{\|Z_{i}\|}{2\theta_{\psi}})+E\sum_{i=1}^{n}\tfrac{E_{i}[\hat{Y}_{i}^{2}]}{\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq B}$
		$\displaystyle\stackrel{{\scriptstyle Lemma~{}\ref{lem:orfnorm-z}}}{{\leq}}n+E\sum_{i=1}^{T-1}\tfrac{\sigma_{i}^{2}}{\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq B}\quad\forall\rho>0.$

Next, we will show the following integral bound:

\displaystyle E\sum_{i=1}^{T-1}\sigma_{i}^{2}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{i-1}^{k}})\mathbbm{1}_{\lambda_{i}^{k}\leq B}\leq\int_{\alpha^{2}}^{(1+\alpha^{2})\wedge B}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{k/2}})\mathrm{d}t.

(55)

Recall the sequence of stopping times $\{\tau_{t}\}_{0\leq t\leq 1}$ defined in (35). Then $\tau_{t}=m$ if and only if $t\in(\sum_{i=1}^{m-1}\sigma_{i}^{2},\sum_{i=1}^{m}\sigma_{i}^{2}]$ . Moreover, recalling that $\beta^{2}=2\alpha^{2}$ ,

\lambda_{\tau_{t}}^{2}=1+\beta^{2}-\sum_{i=1}^{\tau_{t}}\sigma_{i}^{2}\geq 1+\beta^{2}-t-\alpha^{2}=1+\alpha^{2}-t.

Thus, for any $\alpha>0$ , writing $v_{m}^{2}:=\sum_{i=1}^{m}\sigma_{i}^{2}$ ,

	$\displaystyle E\left[\sum_{i=1}^{T-1}\sigma_{i}^{2}\psi_{*}(\tfrac{\alpha^{k}}{\rho\lambda_{i}^{k}})\mathbbm{1}_{\lambda_{i}^{2}\leq B}\right]$	$\displaystyle=E\sum_{i=1}^{T-1}\int_{v_{i-1}^{2}}^{v_{i}^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho\lambda_{\tau_{t}}^{k}})\mathbbm{1}_{\lambda_{\tau_{t}}^{2}\leq B}\mathrm{d}t$
		$\displaystyle\leq E\int_{0}^{1}\psi_{*}(\tfrac{\alpha^{2}}{\rho(1+\alpha^{2}-t)^{k/2}})\mathbbm{1}_{1+\alpha^{2}-t\leq B}\mathrm{d}t$
		$\displaystyle\leq\int_{\alpha^{2}}^{(1+\alpha^{2})\wedge B}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{k/2}})\mathrm{d}t.$

Inequality (55) is proved. The Proposition follows. ∎

To prove Theorem 7 and Theorem 8, we will need the following lemma to compare N-functions to power functions.

Lemma 20.

Let $p>1$ and let $q:=p/(p-1)$ denote its Hölder conjugate. Let $\psi$ be an N-function. Set $C_{\psi}=\max_{x,y>0}\frac{\psi^{-1}(x)\psi_{*}^{-1}(x)/x}{\psi^{-1}(y)\psi_{*}^{-1}(y)/y}$ .

(1)

If $\psi\preccurlyeq x^{p}$ , then

$\frac{\psi_{*}(s)}{s^{q}}\geq C_{\psi}^{-q}\frac{\psi_{*}(t)}{t^{q}}\quad\text{ for any }s>t>0.$
(2)

If $\psi\succcurlyeq x^{p}$ , then

$\frac{\psi_{*}(s)}{s^{q}}\leq C_{\psi}^{q}\frac{\psi_{*}(t)}{t^{q}}\quad\text{ for any }s>t>0.$

Note that $C_{\psi}=1$ if $\psi=x^{p}$ , and (16) guarantees that $1\leq C_{\psi}\leq 2$ in general.

Proof of Theorem 5:.

Without loss of generality, assume $s_{n}=1$ . Set $\theta_{\psi}=\lVert Y\rVert_{\psi}$ .

Part 1 (Proof of Theorem 5(i) for general $\psi$ ).

Since $\psi_{*}(x)/x$ is increasing, we have $t\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\leq\alpha^{2}\psi_{*}(\tfrac{\alpha^{2}}{\rho\alpha^{2}})$ for $t\geq\alpha^{2}$ . Hence

\displaystyle\int_{\alpha^{2}}^{1+\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\mathrm{d}t\leq\int_{\alpha^{2}}^{1+\alpha^{2}}\frac{\alpha^{2}}{t}\psi_{*}(\tfrac{1}{\rho})\mathrm{d}t=\alpha^{2}\psi_{*}(\tfrac{1}{\rho})\log(1+\tfrac{1}{\alpha^{2}}).

This inequality, together with Proposition 19, yields

\displaystyle E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}

\displaystyle\lesssim\theta_{\psi}\rho n+\theta_{\psi}\rho\psi_{*}(\tfrac{1}{\rho})\log(1+\tfrac{1}{\alpha^{2}}).

Taking $\rho=1\big{/}\psi_{*}^{-1}(n)$ in the above inequality, we obtain

\displaystyle E\sum_{i=1}^{T-1}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}\lesssim\theta_{\psi}\rho n\log(e+\tfrac{1}{\alpha^{2}})\lesssim\theta_{\psi}\psi^{-1}(n)\log(e+\tfrac{1}{\alpha^{2}})

(56)

where in the last inequality we used the fact

\rho=1\big{/}\psi_{*}^{-1}(n)\stackrel{{\scriptstyle\eqref{eq:orlicz-ineq}}}{{\leq}}\tfrac{1}{n}\psi^{-1}(n).

(57)

Further, putting

\alpha=2\theta_{\psi}\psi^{-1}(n)=2L_{\psi},

(58)

then inequality (56) and (54) imply

\mathcal{W}_{1}(\hat{X}_{n+1},\mathcal{N})\lesssim\alpha\log(e+\tfrac{1}{L_{\psi}^{2}}).

This inequality, together with Proposition 17(a), yields (recall $K_{1}$ in (42))

\displaystyle\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\alpha\log(e+\tfrac{1}{L_{\psi}^{2}})+\frac{n^{1/2}\alpha}{\psi(\alpha/2\theta_{\psi})^{1/2}}+K_{1}\stackrel{{\scriptstyle\eqref{eq:alpha-value}}}{{\lesssim}}\alpha\log(e+\tfrac{1}{L_{\psi}^{2}})+K_{1}.

Theorem 5(i) is proved.

Part 2 (Proof of Theorem 5(ii) for $\psi\preccurlyeq x^{p}$ , $p>2$ ).

Similar to the previous step, we will first derive a bound for the integral in (55). The key observation is that we can compare $\psi_{*}$ to $(x^{p})_{*}$ instead of $1/x$ (thanks to Lemma 20).

Let $q:=p/(p-1)$ denote the Hölder conjugate of $p>2$ . By Lemma 20, $\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\leq C_{\psi}^{q}(\tfrac{\alpha^{2}}{t})^{q}\psi_{*}(\tfrac{\alpha^{2}}{\rho\alpha^{2}})$ for $t>\alpha^{2}$ . Hence

\int_{\alpha^{2}}^{1+\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\mathrm{d}t\leq\frac{C_{\psi}^{q}}{q-1}\alpha^{2}\psi_{*}(\tfrac{1}{\rho}).

(59)

This inequality, together with Proposition 19, yields (Note that $C_{\psi}^{q}\leq 2^{2}$ .)

\displaystyle E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}

\displaystyle\lesssim\theta_{\psi}\rho n+\tfrac{1}{q-1}\theta_{\psi}\rho\psi_{*}(\tfrac{1}{\rho}).

Putting $\rho=1\big{/}\psi_{*}^{-1}(n)$ , this inequality becomes

\displaystyle E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i-1}^{2}}\lesssim p\theta_{\psi}\rho n\stackrel{{\scriptstyle\eqref{eq:rho-upper}}}{{\lesssim}}p\theta_{\psi}\psi^{-1}(n).

(60)

Hence, taking $\alpha=2\theta_{\psi}\psi^{-1}(n)=2L_{\psi}$ , inequalities (60) and (54) imply

\mathcal{W}_{1}(\hat{X}_{n+1},\mathcal{N})\lesssim p\alpha.

This inequality, together with Proposition 17(a), yields Theorem 5(ii). ∎

4.3 Proof of Theorem 7 ( $\mathcal{W}_{2}$ bounds)

In some sense, our proof of the $\mathcal{W}_{2}$ bound in Theorem 7 is an interpolation between the proofs of Theorem 5 and Theorem 8. We observe that, to bound $E\sum_{i=1}^{n}|Y_{i}|^{3}/\lambda_{i}$ , a better estimate than Proposition 19 can be achieved by simply bounding $|Y_{i}|^{3}/\lambda_{i}$ by $|Y_{i}|^{3}/C$ when $\lambda_{i}$ is larger than some “threshold” $C$ .

Proof of Theorem 7(i):.

Without loss of generality, assume $s_{n}=1$ . Set $\theta_{\psi}=\lVert Y\rVert_{\psi}$ .

Define the mds $\hat{Y}=\hat{Y}^{(\infty)}$ and the martingale $\{\hat{X}_{i}\}$ as in Definition 16. Note that $\hat{Y}_{i}=Y_{i}$ for all $i<T$ .

Let $\beta>0$ be a constant to be determined later, and recall the notations $\sigma_{i},\lambda_{i}$ in (53). Since $\psi\succcurlyeq x^{2}$ and $\lambda_{i}\geq\beta$ for $1\leq i\leq n+1$ , we have $\lambda_{i}^{2}/\psi(\lambda_{i}/\theta_{\psi})\leq\beta^{2}/\psi(\beta/\theta_{\psi})$ for all $i$ . Applying Proposition 13 and Remark 14, we get

	$\displaystyle\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})$	$\displaystyle\lesssim\beta^{2}+E\sum_{i=1}^{n}\frac{\lambda_{i}^{2}\psi(\|\hat{Y}_{i}\|/\theta_{\psi})}{\psi(\lambda_{i}/\theta_{\psi})}$
		$\displaystyle\lesssim\beta^{2}+E\sum_{i=1}^{n}\frac{\beta^{2}\psi(\|Y_{i}\|/\theta_{\psi})}{\psi(\beta/\theta_{\psi})}$
		$\displaystyle\lesssim\beta^{2}+\frac{n\beta^{2}}{\psi(\beta/\theta_{\psi})}.$

Taking $\beta=L_{\psi}=\theta_{\psi}\psi^{-1}(n)$ , we get

\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})\lesssim L_{\psi}^{2}.

Finally, by Proposition 17(b) and (25), with $\alpha=\infty$ ,

\mathcal{W}_{2}(X_{n},\mathcal{N})\lesssim\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})^{1/2}+L_{\psi}+\lVert\sum_{i=1}^{n}\sigma_{i}^{2}-1\rVert_{1}^{1/2}.

Theorem 7(i) follows. ∎

Proof of Theorem 7(ii):.

Without loss of generality, assume $s_{n}=1$ .

Let $\alpha>0$ be a constant to be determined. We define the mds $\hat{Y}=\hat{Y}^{(\alpha)}$ and the martingale $\{\hat{X}_{i}\}$ as in Definition 16. Set $\theta_{\psi}=\lVert Y\rVert_{\psi}$ .

Step 1. Applying Proposition 13 and Remark 14 to the case $\beta=\sqrt{2}\alpha$ , we get

\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})\lesssim\alpha^{2}+E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}.

(61)

Let $k>0$ be a constant to be determined later, and define and event

A_{i}=\{\lambda_{i}\leq k\alpha\}.

By Proposition 19,

E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{\lambda_{i}\leq k\alpha}\lesssim\theta_{\psi}\rho n+\theta_{\psi}\rho\alpha^{-2}\int_{0}^{k^{2}\alpha^{2}}\psi_{*}(\tfrac{\alpha^{2}}{\rho t^{1/2}})\mathrm{d}t

(62)

Since $\psi\succcurlyeq x^{3}$ , by Lemma 20, $t^{3/2}\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\lesssim(k\alpha)^{3/2}\psi_{*}(\tfrac{\alpha^{2}}{\rho k\alpha})$ for $t\in(0,k\alpha)$ . Thus

\displaystyle\int_{0}^{k\alpha}t\psi_{*}(\tfrac{\alpha^{2}}{\rho t})\mathrm{d}t\lesssim(k\alpha)^{3/2}\psi_{*}(\tfrac{\alpha^{2}}{\rho k\alpha})\int_{0}^{k\alpha}t^{1-3/2}\mathrm{d}t\lesssim(k\alpha)^{2}\psi_{*}(\tfrac{\alpha}{\rho k}).

This inequality, together with (62), yields

E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{A_{i}}\lesssim\theta_{\psi}\rho n+\theta_{\psi}\rho k^{2}\psi_{*}(\tfrac{\alpha}{\rho k}).

Taking $\rho$ such that $n=4k^{2}\psi_{*}(\tfrac{\alpha}{\rho k})$ , i.e.,

\rho=\frac{\alpha/k}{\psi_{*}^{-1}(n/k^{2})}\stackrel{{\scriptstyle\eqref{eq:orlicz-ineq}}}{{\leq}}\frac{4\alpha k}{n}\psi^{-1}(\frac{n}{4k^{2}}),

we obtain

E\sum_{i=1}^{T-1}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{A_{i}}\lesssim\theta_{\psi}\rho n\lesssim\theta_{\psi}\alpha k\psi^{-1}(\frac{n}{4k^{2}}).

(63)

Step 2. By Lemma 18, $\lVert\{\hat{Y}_{i}\}_{i=1}^{n}\rVert_{3}^{3}\lesssim\lVert Y\rVert_{3}^{3}$ . Recalling that $A_{i}^{c}=\{\lambda_{i}>k\alpha\}$ and $\psi\succcurlyeq x^{3}$ ,

\displaystyle E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\mathbbm{1}_{A_{i}^{c}}\leq E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{k\alpha}\lesssim\frac{n\lVert Y\rVert_{3}^{3}}{k\alpha}\lesssim_{\psi(1)}\frac{n\theta_{\psi}^{3}}{k\alpha}.

where Lemma A.1 is used in the last inequality. This inequality, together with (63), yields

E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\lesssim_{\psi(1)}\theta_{\psi}\alpha k\psi^{-1}(\frac{n}{4k^{2}})+\frac{n\theta_{\psi}^{3}}{k\alpha}.

Let $g(x):=\frac{1}{x}\psi(x)$ . Choosing $k$ such that $\theta_{\psi}\alpha k\psi^{-1}(\frac{n}{4k^{2}})=\frac{n\theta_{\psi}^{3}}{k\alpha}$ , i.e.,

\frac{1}{\alpha k}=\sqrt{g^{-1}(\tfrac{\alpha^{2}}{4\theta_{\psi}^{2}})/(n\theta_{\psi}^{2})},

we arrive at the bound

E\sum_{i=1}^{n}\frac{|\hat{Y}_{i}|^{3}}{\lambda_{i}}\lesssim_{\psi(1)}\theta_{\psi}^{2}\sqrt{ng^{-1}(\tfrac{\alpha^{2}}{4\theta_{\psi}^{2}})}.

Step 3. Further, by (61), we get

\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})\lesssim_{\psi(1)}\alpha^{2}+\theta_{\psi}^{2}\sqrt{ng^{-1}(\tfrac{\alpha^{2}}{\theta_{\psi}^{2}})}\,.

Thus, choosing $\alpha=2L_{\psi}=2\theta_{\psi}\psi^{-1}(n)$ , by Proposition 17(a) and (25),

	$\displaystyle\mathcal{W}_{2}(X_{n},\mathcal{N})$	$\displaystyle\lesssim\mathcal{Z}_{2}(\hat{X}_{n+1},\mathcal{N})^{1/2}+L_{\psi}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}$
		$\displaystyle\lesssim_{\psi(1)}L_{\psi}+\theta_{\psi}\left[ng^{-1}(\psi^{-1}(n)^{2})\right]^{1/4}+\lVert V_{n}^{2}-1\rVert_{1}^{1/2}.$

Theorem 7(ii) is proved. ∎

5 Optimality of the $\mathcal{W}_{1}$ rates: Proof of Proposition 10

For any $p>2$ , we will construct a mds $\{Y_{i}\}_{i=1}^{n}$ such that $V_{n}^{2}=1$ and $\mathcal{W}_{1}(X_{n},\mathcal{N})\gtrsim_{p}L_{p}(Y)$ . Note that in our examples, $L_{p}(Y)\to 0$ as $n\to\infty$ , justifying the optimality of the term $L_{\psi}$ in Theorem 5.

We choose

\alpha=\frac{1}{\log n}.

(Actually for any $p>2$ , any $\alpha\gg n^{1/p-1/2}$ so that $\alpha_{n}\to 0$ as $n\to\infty$ would work.)

Example 21.

For $n\geq 3$ , let $Y_{1},\ldots,Y_{n-2},Y_{n},\xi,\eta$ be independent random variables such that

•

$Y_{n},\xi\sim\mathcal{N}(0,\alpha_{n}^{2})$ ,
•

$Y_{1},\ldots,Y_{n-2}$ are i.i.d. $\mathcal{N}(0,\tfrac{1-2\alpha_{n}^{2}}{n-2})$ normal random variables,
•

$\eta$ has distribution

$P(\eta=\tfrac{1}{2})=\tfrac{4}{5},\quad P(\eta=-2)=\tfrac{1}{5},$

We let $X_{m}:=Y_{1}+\ldots+Y_{m}$ and define a set $A\subset\mathbb{R}$ as

A=\left\{x\in\mathbb{R}:\cos x\geq\tfrac{1}{2}\right\}=\bigcup_{k\in\mathbb{Z}}\big{[}(2k-\tfrac{1}{3})\pi,(2k+\tfrac{1}{3})\pi\big{]}.

Define

Y_{n-1}=\xi\mathbbm{1}_{X_{n-2}\notin\alpha A}+\alpha\eta\mathbbm{1}_{X_{n-2}\in\alpha A}.

Clearly, $\{X_{m}\}_{m=1}^{n}$ is a martingale, and

\sigma_{n-1}^{2}=\sigma_{n}^{2}=\alpha^{2},\quad\sigma_{i}^{2}=\tfrac{1-2\alpha^{2}}{n-2}\,\text{ for }i=1,\ldots,n-2.

Of course, $V_{n}^{2}=\sum_{i=1}^{n}\sigma_{i}^{2}=1$ almost surely, and

L_{p}=\left(\sum_{i=1}^{n}E[|Y_{i}|^{p}]\right)^{1/p}\leq\left((n-2)\frac{c}{n^{p/2}}+c\alpha^{p}\right)^{1/p}\lesssim_{p}\alpha.

Proof of Proposition 10(1).

Let $\{Y_{i}\}_{i=1}^{n}$ be the mds in Example 21. We consider the function

h(x)=\alpha\sin(\tfrac{x}{\alpha}).

Clearly, $[h]_{0,1}\leq 1$ . Recall the definition of the function $h_{\alpha}(x)$ in (26). Then

h_{\alpha}(x)=\alpha\int_{\mathbb{R}}\sin\big{(}\tfrac{1}{\alpha}(x-\alpha u)\big{)}\phi(u)\mathrm{d}u=h(x)\int_{\mathbb{R}}\cos u\phi(u)\mathrm{d}u=\frac{1}{\sqrt{e}}h(x),

(64)

where $\phi(x)=\tfrac{1}{\sqrt{2\pi}}e^{-x^{2}/2}$ denotes the standard normal density. Moreover,

\displaystyle E[h(X_{n})-h(\mathcal{N})]=E[h_{\alpha}(X_{n-1})-h_{\alpha}(X_{n-2}+\alpha\mathcal{N})].

By the definition of $Y_{n-1}$ , we have

\displaystyle E[h_{\alpha}(X_{n-1})]=E[h_{\alpha}(X_{n-2}+\alpha\mathcal{N})\mathbbm{1}_{X_{n-2}\notin\alpha A}]+E[h_{\alpha}(X_{n-1}+\alpha\eta)\mathbbm{1}_{X_{n-2}\in\alpha A}].

Thus

		$\displaystyle E[h(X_{n})-h(\mathcal{N})]$		(65)
		$\displaystyle=E\left[\left(h_{\alpha}(X_{n-2}+\alpha\eta)-h_{\alpha}(X_{n-2}+\alpha\mathcal{N})\right)\mathbbm{1}_{X_{n-1}\in\alpha A}\right]$
		$\displaystyle\stackrel{{\scriptstyle\eqref{eq:calcul-h}}}{{=}}\tfrac{\alpha}{\sqrt{e}}E\left[\left(\sin\tfrac{X_{n-2}}{\alpha}(\cos\eta-\cos\mathcal{N})+\cos\tfrac{X_{n-2}}{\alpha}(\sin\eta-\sin\mathcal{N})\right)\mathbbm{1}_{X_{n-2}\in\alpha A}\right]$
		$\displaystyle=\tfrac{\alpha}{\sqrt{e}}\left(E\left[\sin\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha A}\right]E[\cos\eta-\cos\mathcal{N}]+E\left[\cos\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha A}\right]E[\sin\eta]\right).$

Since the set $A$ is symmetric, i.e., $A=-A$ , and $X_{n-2}\sim\mathcal{N}(0,1-2\alpha^{2})$ , we get $E\left[\sin\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha A}\right]=0$ . Hence

	$\displaystyle E[h(X_{n})-h(\mathcal{N})]$	$\displaystyle=\tfrac{\alpha}{\sqrt{e}}E\left[\cos\tfrac{X_{n-2}}{\alpha}\mathbbm{1}_{X_{n-2}\in\alpha A}\right]E[\sin\eta]$
		$\displaystyle\geq\tfrac{\alpha}{2\sqrt{e}}P(X_{n-2}\in\alpha A)E[\sin\eta].$

Note that $E[\sin\eta]=\tfrac{4}{5}\sin\tfrac{1}{2}-\tfrac{1}{5}\sin 2>0.2$ , and, writing $c_{1}:=(1-2\alpha^{2})^{-1/2}$ ,

$\displaystyle P(X_{n-2}\in\alpha A)$	$\displaystyle=P(\mathcal{N}\in c_{1}\alpha A)$
	$\displaystyle=\sum_{k\in\mathbb{Z}}\int_{c_{1}(6k-1)\alpha\pi/3}^{c_{1}(6k+1)\alpha\pi/3}\phi(u)\mathrm{d}u$
	$\displaystyle\geq\frac{1}{2}\left(\int_{0}^{c_{1}\alpha\pi/3}\phi(u)\mathrm{d}u+\sum_{k=0}^{\infty}\tfrac{2c_{1}\alpha\pi}{3}\phi(\tfrac{c_{1}(6k+1)\alpha\pi}{3})\right)$
	$\displaystyle\geq\frac{1}{6}\int_{0}^{\infty}\phi(u)\mathrm{d}u=\frac{1}{12}.$	(66)

Therefore, $E[h(X_{n})-h(\mathcal{N})]\geq\frac{\alpha}{120\sqrt{e}}$ and so

\mathcal{W}_{1}(X_{n},\mathcal{N})\geq\frac{\alpha}{120\sqrt{e}}\geq CL_{p},

justifying the optimality of the power of $L_{\psi}$ in Theorem 5. ∎

We can modify Example 21 above to justify the optimality of the exponent of $\lVert V^{2}-1\rVert_{1/2}$ in Theorem 5.

Example 22.

For $n\geq 3$ , let $Y_{1},\ldots,Y_{n-2}$ be i.i.d. with $Y_{1}\sim\mathcal{N}(0,\frac{1-\alpha^{2}}{n-2})$ . Let

Y_{n-1}=\alpha\eta\mathbbm{1}_{X_{n-2}\in\alpha A},

where $X_{n-2}=\sum_{i=1}^{n-1}Y_{i}$ , $\alpha=\alpha_{n}=1/\log n$ , $\eta$ and the set $A\subset\mathbb{R}$ are as in Example 21. Let $Y_{n}\sim\mathcal{N}\left(0,\alpha^{2}P(\mathcal{N}\notin\kappa A)\right)$ , where $\kappa=\kappa_{n}=\alpha/\sqrt{1-\alpha^{2}}$ .

Proof of Proposition 10(2).

Let the mds $\{Y_{i}\}_{i=1}^{n}$ be as in Example 22.

Clearly, $L_{3}=\left(\sum_{i=1}^{n}E[|Y_{i}|^{3}]\right)^{1/3}\lesssim\alpha$ , and

	$\displaystyle\lVert V_{n}^{2}-1\rVert_{1/2}$	$\displaystyle=\alpha^{2}E\left[\|\mathbbm{1}_{X_{n-2}\in\alpha A}-P(X_{n-2}\in\alpha A)\|^{1/2}\right]^{2}$
		$\displaystyle\asymp\alpha^{2}P(\mathcal{N}\notin\kappa A)P(\mathcal{N}\in\kappa A).$

Hence, by Theorem 5(ii), we get

\mathcal{W}_{1}(X_{n},\mathcal{N})\lesssim\alpha,

(67)

Next we will show that, there exists $N>0$ such that for $n>N$ ,

\lVert V_{n}^{2}-1\rVert_{1/2}\asymp\alpha^{2}.

(68)

Note that the same computation as in (5) yields $P(\mathcal{N}\in\kappa A)\geq 1/12$ . Hence we only need to show that $P(\mathcal{N}\notin\kappa A)\gtrsim 1$ . Indeed,

	$\displaystyle P(\mathcal{N}\in\kappa A)$	$\displaystyle=\sum_{i\in\mathbb{Z}}\int_{c_{1}(6i-1)\kappa\pi/3}^{c_{1}(6i+1)\kappa\pi/3}\phi(u)\mathrm{d}u$
		$\displaystyle\leq 2\int_{0}^{c_{1}\kappa\pi/3}\phi(u)\mathrm{d}u+2\sum_{i=1}^{\infty}\frac{2c_{1}\kappa\pi}{3}\phi(\frac{c_{1}(6i-1)\kappa\pi}{3})$
		$\displaystyle\leq 2\int_{0}^{c_{1}\kappa\pi/3}\phi(u)\mathrm{d}u+\frac{2}{3}\int_{-c_{1}\kappa\pi/3}^{\infty}\phi(u)\mathrm{d}u$
		$\displaystyle\leq\frac{8}{3}\int_{0}^{c_{1}\kappa\pi/3}\phi(u)\mathrm{d}u+\frac{1}{3}\leq\frac{1}{2}$

for all $n$ sufficiently big. Inequality (68) follows.

It remains to show that

\mathcal{W}_{1}(X_{n},\mathcal{N})\gtrsim\alpha.

To this end, we consider the function $h(x)=\alpha\sin(\tfrac{x}{\alpha})$ . Define $\lambda>0$ by

\lambda^{2}={\rm Var}(Y_{n})=\alpha^{2}P(\mathcal{N}\notin\kappa A)\in[\tfrac{1}{2}\alpha^{2},\tfrac{11}{12}\alpha^{2}].

Recall the definition of $h_{\lambda}(x)$ in (26). By the same calculation as in (64),

h_{\lambda}(x)=h(x)\int_{\mathbb{R}}\cos(\tfrac{\lambda}{\alpha}u)\phi(u)\mathrm{d}u=h(x)\exp(-\frac{\lambda^{2}}{2\alpha^{2}}).

By symmetry, $E[h(\mathcal{N})]=0$ and $E[h_{\lambda}(X_{n-2})\mathbbm{1}_{X_{n-2}\notin\alpha A}]=0$ . Hence

	$\displaystyle E[h(X_{n})-h(\mathcal{N})]$
	$\displaystyle=E[h_{\lambda}(X_{n-2})\mathbbm{1}_{X_{n-2}\notin\alpha A}]+E[h_{\lambda}(X_{n-2}+\alpha\eta)\mathbbm{1}_{X_{n-2}\in\alpha A}]$
	$\displaystyle=\alpha\exp(-\frac{\lambda^{2}}{2\alpha^{2}})E[\cos(\tfrac{X_{n-2}}{\alpha})\sin\eta\mathbbm{1}_{X_{n-2}\in\alpha A}]$
	$\displaystyle\geq 0.1\alpha\exp(-\frac{\lambda^{2}}{2\alpha^{2}})P(X_{n-2}\in\alpha A)\geq c\alpha.$

Display (67) is proved. Our proof of the Proposition is complete. ∎

6 Some open questions

1.

For the $\mathscr{L}^{\infty}$ case (i.e. the mds is uniformly $\mathscr{L}^{\infty}$ -bounded), is the typical $\mathcal{W}_{1}$ rates $O(n^{-1/2}\log n)$ within Theorem 5 optimal? Recall that it is shown by Bolthausen [5] that this rate is typically optimal for the Kolmogorov distance.
2.

For the $\mathscr{L}^{\infty}$ case, are the typical $\mathcal{W}_{r}$ rates $O(n^{-/(2r)})$ , $r\in\{2,3\}$ , in Theorems 7 and 8 optimal? For general $r>3$ and bounded mds, can we get similar bounds $O(n^{-/(2r)})$ , $r>3$ ?
3.

Can we say anything about the optimality of the $\mathcal{W}_{2}$ rates in Theorem 7 when the mds is $\mathscr{L}^{p}$ integrable, for any $p>2$ ?
4.

Is there a better (unified) formula than the “piecewise” $\mathcal{W}_{2}$ bound in Theorem 7 for $\mathscr{L}^{p}$ martingales, $p>2$ ?
5.

As a feature of the $\mathcal{W}_{1},\mathcal{W}_{2}$ bounds in Theorems 5 and 7, better integrability implies faster (typical) convergence rates for the martingale CLT. But this is no longer the case for the $\mathcal{W}_{3}$ estimate in Theorem 8. Is it possible to get better $\mathcal{W}_{3}$ bounds for general martingales with better integrability than $\mathscr{L}^{3}$ ?
6.

How to obtain the Wasserstein rates for the CLT of multi-dimensional martingales? Can we say something on the dependence of the rates on the dimension?
7.

How to obtain the Wasserstein- $r$ convergence rates, $r>3$ , for the martingale CLT in terms of $L_{\psi}$ ?

Appendix A Appendix

A.1 Comparison between Orlicz norms

Recall the definition of the (mean-)orlicz norm $\lVert\cdot\rVert_{\psi}$ for a sequence in Definition 3.

Lemma A.1.

Let $p\geq 1$ and let $Y=\{Y_{i}\}_{i=1}^{n}$ be a sequence of random variables with $\lVert Y\rVert_{p}<\infty$ . If an N-function $\psi$ satisfies $\psi\succcurlyeq x^{p}$ , then

\lVert Y\rVert_{p}\leq\left(1+\frac{1}{\psi(1)}\right)^{1/p}\lVert Y\rVert_{\psi}.

Proof.

Without loss of generality assume $\lVert Y\rVert_{\psi}=1$ . Since $\psi/x^{p}$ is increasing,

\displaystyle E[|Y_{i}|^{p}]\leq 1+E[|Y_{i}|^{p}\mathbbm{1}_{|Y_{i}|>1}]\leq 1+\frac{1}{\psi(1)}E\left[\psi(|Y_{i}|)\right].

Thus, by the definition of $\lVert\cdot\rVert_{\psi}$ ,

\displaystyle\lVert Y\rVert_{p}^{p}=\frac{1}{n}\sum_{i=1}^{n}E[|Y_{i}|^{p}]\leq 1+\frac{1}{\psi(1)n}\sum_{i=1}^{n}E[\psi(|Y_{i}|)]\leq 1+\frac{1}{\psi(1)}.

The lemma follows. ∎

A.2 Regularity of Gaussian smoothing:Proof of Lemma 12

The proof is exactly as in [10, Lemma 6.1].

Proof.

Since $f_{\sigma}(x)=\int_{\mathbb{R}}f(x-\sigma u)\phi(u)$ , integration by parts yields

	$\displaystyle f_{\sigma}^{(k)}(x)$	$\displaystyle=\frac{1}{\sigma^{k}}\int_{\mathbb{R}}f^{(\ell)}(x-\sigma u)\phi^{(k-\ell)}(u)\mathrm{d}u$
		$\displaystyle=\frac{1}{\sigma^{k-\ell}}\int_{\mathbb{R}}[f^{(\ell)}(x-\sigma u)-f^{(\ell)}(x)]\phi^{(k-\ell)}(u)\mathrm{d}u,\quad\forall 0\leq\ell<k.$

The lemma follows by taking $\ell=r-1$ and using the fact $[f]_{r-1,1}=1$ . ∎

A.3 Moment bound of the maximum

Lemma A.2.

Let $\psi\succcurlyeq x^{r}$ be an N-function, $r>0$ . For any sequence of random variables $Y=\{Y_{i}\}_{i=1}^{n}$ , we have

\lVert\max_{i=1}^{n}|Y_{i}|\rVert_{r}\leq 2^{1/r}\lVert Y\rVert_{\psi}\psi^{-1}(n).

Proof.

Without loss of generality, assume $\lVert Y\rVert_{\psi}=1$ . For any $\alpha>0$ ,

|Y_{i}|^{r}\mathbbm{1}_{|Y_{i}|\geq\alpha}\leq\frac{\alpha^{r}}{\psi(\alpha)}\psi(|Y_{i}|)\mathbbm{1}_{|Y_{i}|\geq\alpha},\quad 1\leq i\leq n

where we used the fact that $x\mapsto\frac{\psi}{x^{r}}$ is increasing. Hence

	$\displaystyle E[\max_{i=1}^{n}\|Y_{i}\|^{r}]$	$\displaystyle\leq\alpha^{r}+E[\max_{i=1}^{n}\|Y_{i}\|^{r}\mathbbm{1}_{\|Y_{i}\|\geq\alpha}]$
		$\displaystyle\leq\alpha^{r}+E\left[\sum_{i=1}^{n}\frac{\alpha^{r}}{\psi(\alpha)}\psi(\|Y_{i}\|)\right]\leq\alpha^{r}+\frac{n\alpha^{r}}{\psi(\alpha)}$

Taking $\alpha=\psi^{-1}(n)$ , we get $\lVert\max_{i=1}^{n}|Y_{i}|\rVert_{r}^{r}\leq 2\alpha^{r}$ . ∎

A.4 Proof of Lemma 20

Proof.

We only give the proof of (1). The proof of (2) is similar.

The case $\psi=x^{p}$ is trivial, so we only consider the general case $\psi\preccurlyeq x^{p}$ . It suffices to show that, for any $s>t>0$ ,

\frac{s^{1/q}}{\psi_{*}^{-1}(s)}\geq\frac{1}{4}\frac{t^{1/q}}{\psi_{*}^{-1}(t)},

which, writing $F(x):=\psi^{-1}(x)\psi_{*}^{-1}(x)$ , is equivalent to

\frac{\psi^{-1}(s)/s^{1/p}}{\psi^{-1}(t)/t^{1/p}}\geq\frac{1}{C_{\psi}}\frac{F(s)/s}{F(t)/t}\quad\forall s>t>0.

Note that $\psi\preccurlyeq x^{p}$ implies that the left-side is at least 1. Lemma 20(1) follows. ∎

A.5 A Burkholder inequality

Theorem A.3.

Let $\{Y_{i}\}_{i=1}^{n}$ be a mds and let $S_{m}=Y_{1}+\ldots+Y_{m}$ , $1\leq m\leq n$ . Recall the notation $\sigma_{i}(Y)^{2}$ in (1). Then, fir $p>0$ ,

E[\max_{i=1}^{n}|S_{n}|^{p}]\lesssim_{p}E[\big{(}\sum_{i=1}^{n}\sigma_{i}^{2}(Y)\big{)}^{p/2}]+E[\max_{i=1}^{n}|Y_{i}|^{p}].

This version is taken from [27, Theorem 2.11]. For a more general inequality, see [7, Theorem 21.1].

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E-mail address, Xiaoqin Guo: [email protected]

	$\displaystyle E_{i}[\psi(\|Z_{i}\|/2)]$	$\displaystyle\leq E_{i}\left[\psi\left((\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2}+E_{i}[\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2}])/2\right)\right]$
		$\displaystyle\leq\frac{1}{2}E_{i}\left[\psi(\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2})+\psi(E_{i}[\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2}])\right]$
		$\displaystyle\leq E_{i}\left[\psi(\|Y_{i}\|\mathbbm{1}_{\|Y_{i}\|\leq\alpha/2})\right],$

On the rate of convergence of the martingale central limit theorem in Wasserstein distances

Abstract

1 Introduction

1.1 Earlier results in the literature

Remark 1.

1.2 Motivation and our contributions

1.3 Structure of the paper

1.4 Preliminaries: N-functions and an Orlicz-norm for sequences

Definition 2.

Definition 3.

Remark 4.

2 Main results

Theorem 5 (𝒲1\mathcal{W}_{1} bounds).

Remark 6.

Theorem 7 (𝒲2\mathcal{W}_{2} bounds).

Theorem 8 (𝒲3\mathcal{W}_{3} bound).

Remark 9.

Proposition 10.

3 Martingales with Vn2=1V_{n}^{2}=1

3.1 Wasserstein distance bounds via Lindeberg’s argument

Definition 11.

Lemma 12.

Proposition 13.

Proof of Proposition 13:.

Remark 14.

3.2 The rates of some special cases

Corollary 15.

Proof.

4 Proofs of the main 𝒲r\mathcal{W}_{r} bounds

4.1 A modified martingale, and Proof of Theorem 8 (𝒲3\mathcal{W}_{3} bounds)

Definition 16.

Proposition 17.

Lemma 18.

Proof.

Proof of Proposition 17.

Proof of Theorem 8.

4.2 Proof of Theorem 5 (𝒲1\mathcal{W}_{1} bounds)

Proposition 19.

Proof.

Lemma 20.

Proof of Theorem 5:.

4.3 Proof of Theorem 7 (𝒲2\mathcal{W}_{2} bounds)

Proof of Theorem 7(i):.

Proof of Theorem 7(ii):.

5 Optimality of the 𝒲1\mathcal{W}_{1} rates: Proof of Proposition 10

Example 21.

Proof of Proposition 10(1).

Example 22.

Proof of Proposition 10(2).

6 Some open questions

Appendix A Appendix

A.1 Comparison between Orlicz norms

Lemma A.1.

Proof.

A.2 Regularity of Gaussian smoothing:Proof of Lemma 12

Proof.

A.3 Moment bound of the maximum

Lemma A.2.

Proof.

A.4 Proof of Lemma 20

Proof.

A.5 A Burkholder inequality

Theorem A.3.

References

Theorem 5 ( $\mathcal{W}_{1}$ bounds).

Theorem 7 ( $\mathcal{W}_{2}$ bounds).

Theorem 8 ( $\mathcal{W}_{3}$ bound).

3 Martingales with $V_{n}^{2}=1$

4 Proofs of the main $\mathcal{W}_{r}$ bounds

4.1 A modified martingale, and Proof of Theorem 8 ( $\mathcal{W}_{3}$ bounds)

4.2 Proof of Theorem 5 ( $\mathcal{W}_{1}$ bounds)

4.3 Proof of Theorem 7 ( $\mathcal{W}_{2}$ bounds)

5 Optimality of the $\mathcal{W}_{1}$ rates: Proof of Proposition 10