Convergence of persistence diagrams
for discrete time stationary processes

Andrew M. Thomas Department of Statistics and Actuarial Science, University of Iowa [email protected]

Abstract.

In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a measure “above the diagonal” in the extended plane) evaluated on a large class of sets and functions—more than just continuous functions with compact support. We prove this result subject to only minor conditions that the sequence is ergodic and the tails of the marginals are not too heavy. The second result is a central limit theorem for the persistence diagram evaluated on the class of all step functions; this result holds as long as a $\rho$ -mixing criterion is satisfied and the distributions of the partial maxima do not decay too slowly. Our results greatly expand those extant in the literature to allow for more fruitful use in statistical applications, beyond idealized settings. Examples of distributions and functions for which the limit theory holds are provided throughout.

A portion of this work was completed while the author was a postdoctoral associate at Cornell University. This work was funded in part by NSF grants DMS-2114143 and OAC-1940124.

1. Introduction

Understanding the persistent homology of large samples from various probability distributions is of increasing utility in goodness-of-fit testing (Biscio et al., 2020; Krebs and Hirsch, 2022). For goodness-of-fit testing in the “geometric” setting there are a number of results to choose from, as much attention has been focused on the limiting stochastic behavior of Čech and Vietoris-Rips persistent homology of (Euclidean) point clouds (ibid. as well as Hiraoka et al., 2018; Divol and Polonik, 2019; Krebs and Polonik, 2019; Owada and Bobrowski, 2020; Krebs, 2021; Owada, 2022; Bobrowski and Skraba, 2024). However, less attention has been focused on the asymptotics of the entire sublevel (or superlevel) set persistent homology of stochastic processes and random fields—with a few notable exceptions (Chazal and Divol, 2018; Baryshnikov, 2019; Miyanaga, 2023; Perez, 2023; Kanazawa et al., 2024).

In recent years, summaries of sublevel set persistent homology of time series—such as those we establish limit theory for below—have been applied to the problems of heart rate variability analysis (Chung et al., 2021; Graff et al., 2021), eating behavior detection (Chung et al., 2022), and sleep stage scoring using respiratory signals (Chung et al., 2024). Thus, a comprehensive treatment of the asymptotic properties of sublevel set persistent homology of stochastic processes is needed for rigorous statistical approaches to the aformentioned problems. In this article we greatly extend the existing limit theory for persistence diagrams derived from sublevel set filtrations of discrete time stochastic processes. As a result, we understand the behavior of certain real-valued summaries of these random persistence diagrams—so-called persistence statistics—that are particularly relevant to machine learning and goodness-of-fit testing.

Work pertaining to the topology of sub/superlevel sets of random functions has its most prominent originator in Rice (1944). Current work in the area of establishing results about the sublevel set ( $0^{th}$ ) persistent homology of stochastic processes has focused on almost surely continuous processes, such as investigations into the expected persistence diagrams of Brownian motion (Chazal and Divol, 2018); expected persistence diagrams of Brownian motion with drift (Baryshnikov, 2019); and expectations for the number of barcodes and persistent Betti numbers $\beta^{s,t}_{0}$ of continuous semimartingales (Perez, 2023). The formulas in Perez (2023), save for the expected number of barcodes with lifetime greater than $\ell$ , follow asymptotic formulas with $\ell$ tending to 0 or $\infty$ .

Though not overlapping entirely with our setting, some results for cubical persistent homology are applicable here. Notable results include the strong law of large numbers for persistence diagrams (Kanazawa et al., 2024) of random cubical sets (with the quality of the strong law being vague convergence) and central limit theorems for persistent betti numbers of sublevel sets of i.i.d. sequences found in Miyanaga (2023). In this article, we establish the most general strong law of large numbers yet for functionals of persistence diagrams. We do so by normalizing the persistence diagrams so they become probability measures and by leveraging the tools of weak convergence. We also prove a central limit theorem for persistence diagrams evaluated on step functions using recent results for weakly dependent and potentially nonstationary triangular arrays, subject to standard dependence decay conditions on the underlying stationary sequence.

The quality of most strong laws of large numbers for persistence diagrams has been vague convergence, with Hiraoka et al. (2018), Krebs (2021), and Owada (2022) tackling the geometric (i.e. Čech and Vietoris-Rips persistent homology) setting, and Kanazawa et al. (2024) addressing the cubical setting. Recently however, the authors of Bobrowski and Skraba (2024) have employed the weak convergence ideas that we use here to prove a strong law of large numbers for the probability measure defined by death/birth ratios in a persistence diagram, for the geometric setting. In Divol and Polonik (2019)—again in the geometric setting—the authors extend the set functions for which the strong law of Hiraoka et al. (2018) holds to a class of unbounded functions.

In Section 3.1, we accomplish this extension as well in the setting of sublevel set persistent homology. We extend the strong law of large numbers (SLLN) of Kanazawa et al. (2024) (that which pertains to the 1-dimensional setting) from continuous functions with compact support to a large class of unbounded functions. We achieve this based solely on minor conditions such as ergodicity and restrictions on the heaviness of the tails of the marginal distributions of our underlying stochastic process. We also remove the need for any local dependence condition, such as that of Kanazawa et al. (2024). In doing so, we answer an open question of Chung et al. (2021) about the limiting empirical distribution of persistence diagram lifetimes for sublevel sets of discrete time stationary processes. For this specific setting, we also derive an explicit representation of the strong limit of our sublevel set persistent betti numbers in Proposition 3.3, answering a query set forth in the conclusion to Hiraoka and Tsunoda (2018). Finally, we extend the current state-of-the art result central limit theorem (CLT) for persistent Betti numbers of sublevel set filtrations of 1-dimensional processes (Theorem 1.2.3 in Miyanaga, 2023) to finite-dimensional convergence and beyond the realm of i.i.d. observations.

This article proceeds in Section 2 with a treatment of persistent homology specialized to our setting, as well as details of our probabilistic setup. In Section 3 the strong law of large numbers is stated and proved (Theorems 3.1 and 3.8, on pages 3.1 and 3.8) and examples for which it holds are given for specific unbounded functionals of persistence diagrams in Corollary 3.10. Beyond this, we derive some satisfying results in the case of i.i.d. stochastic processes in Corollary 3.5 and state a Glivenko-Cantelli theorem for persistence lifetimes in Corollary 3.7. Finally in Section 4, we state the setting and results of our central limit theorem for persistence diagrams (Theorem 4.6, on page 4.6). We conclude with a brief discussion about the potential improvements and extensions of this work in Section 5. The proof of the central limit theorem is deferred to Section 6.

2. Background

We begin by discussing the necessary notions in topological data analysis—specifically zero-dimensional sublevel set persistent homology. From there, we detail crucial results for the representation of zero-dimensional sublevel set persistent homology for stochastic processes.

Before continuing, let us make a brief note about notation. For a real numbers $x,y$ we define $x\wedge y\mathrel{\mathop{\mathchar 58\relax}}=\min\{x,y\}$ , $x\vee y\mathrel{\mathop{\mathchar 58\relax}}=\max\{x,y\}$ , and $(x)_{+}\mathrel{\mathop{\mathchar 58\relax}}=x\vee 0=\max\{x,0\}$ . We set $\bar{\mathbb{R}}\mathrel{\mathop{\mathchar 58\relax}}=[-\infty,\infty]$ and $\mathbb{R}_{+}\mathrel{\mathop{\mathchar 58\relax}}=[0,\infty)$ . If $R$ is a set in some topological space, we denote $R^{\circ}$ the interior (i.e. largest open subset) of $R$ and $\partial R$ its boundary. We denote $B(z,\epsilon)$ to be the open Euclidean ball of radius $\epsilon>0$ centered at $z$ . If for a real sequence $(a_{n})_{n\geq 1}$ and a positive sequence $(b_{n})_{n\geq 1}$ we have $a_{n}/b_{n}\to 0$ as $n\to\infty$ , we write $a_{n}=o(b_{n})$ ; if there exists a $C>0$ such that $|a_{n}|\leq Cb_{n}$ for $n$ large enough, we write $a_{n}=O(b_{n})$ .

2.1. Homology

Recall that an (abstract) simplicial complex $K$ is a collection of subsets of a set $A$ with the property that it is closed under inclusion. Let $K$ be the graph (i.e. a special case of a simplicial complex) with vertex set $V=\{v_{0},v_{1},v_{2},\dots\}$ and edge set

\{v_{0}v_{1},v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},\dots\}.

For a fixed function $f\mathrel{\mathop{\mathchar 58\relax}}K\to\mathbb{R}$ that satisfies $\tau\subset\sigma$ $\Rightarrow$ $f(\tau)\leq f(\sigma)$ , we define $K(t)\mathrel{\mathop{\mathchar 58\relax}}=\{\sigma\in K\mathrel{\mathop{\mathchar 58\relax}}f(\sigma)\leq t\}$ . It is clear that for $s\leq t$ we have $K(s)\subset K(t)$ and thus $K=\big{(}K(t)\big{)}_{t\in\mathbb{R}}$ defines a filtration of graphs. For any $t\in\mathbb{R}$ we can assess the connectivity information of $K(t)$ by calculating its $0$ -dimensional homology group $H_{0}(K(t))$ . We do so by initially forming two vector spaces $C_{0}$ and $C_{1}$ of all formal linear combinations of the vertices

C_{0}(K(t))\mathrel{\mathop{\mathchar 58\relax}}=\Bigg{\{}\sum_{i\mathrel{\mathop{\mathchar 58\relax}}\,v_{i}\in K(t)}a_{i}v_{i}\mathrel{\mathop{\mathchar 58\relax}}a_{i}\in\mathbb{Z}_{2}\Bigg{\}}

and

C_{1}(K(t))\mathrel{\mathop{\mathchar 58\relax}}=\Bigg{\{}\sum_{i\mathrel{\mathop{\mathchar 58\relax}}\,v_{i}v_{i+1}\in K(t)}a_{i}v_{i}v_{i+1}\mathrel{\mathop{\mathchar 58\relax}}a_{i}\in\mathbb{Z}_{2}\Bigg{\}},

where $\mathbb{Z}_{2}$ is the field of two elements $\{0,1\}$ . The elements of $C_{0}(K(t))$ and $C_{1}(K(t))$ are called 0-chains and 1-chains, respectively. Addition of $i$ -chains in $C_{i}(K(t))$ is done componentwise. To calculate $H_{0}(K(t))$ we need to specify the boundary map $\partial_{1}\mathrel{\mathop{\mathchar 58\relax}}C_{1}(K(t))\to C_{0}(K(t))$ , which is defined by

\partial_{1}\big{(}v_{i}v_{i+1}\big{)}=v_{i}+v_{i+1}.

We can extend this to an arbitrary $c\in C_{1}(K(t))$ by

\partial_{1}(c)=\sum_{i\mathrel{\mathop{\mathchar 58\relax}}\,v_{i}v_{i+1}\in K(t)}a_{i}\partial_{1}(v_{i}v_{i+1}).

By analogy to the construction above, each vertex in $C_{0}(K(t))$ gets sent to 0 by $\partial_{0}$ so $Z_{0}(K(t))\mathrel{\mathop{\mathchar 58\relax}}=\ker\partial_{0}=\#\{v\in C_{0}(K(t))\}$ . Defining $B_{0}(K(t))\mathrel{\mathop{\mathchar 58\relax}}=\partial_{1}\big{(}C_{1}(K(t))\big{)}$ (the image of $\partial_{1}$ ), we define the $0^{th}$ homology group as the quotient vector space,

H_{0}(K(t))\mathrel{\mathop{\mathchar 58\relax}}=Z_{0}(K(t))/B_{0}(K(t)).

A more general setup of homology with $\mathbb{Z}_{2}$ coefficients can be seen in Chapter 4 of Edelsbrunner and Harer (2010).

2.2. Persistent homology and representations

The vector spaces¹¹1Conventionally called groups, as coefficients may lie in $\mathbb{Z}$ , for example. $H_{0}(K(t))$ capture intuitive connectivity information—the elements of $H_{0}(K(t))$ are the equivalence classes of vertices that satisfy $v+v^{\prime}\in B_{0}(K(t))$ . More simply put, elements of $H_{0}(K(t))$ are vertices connected by a chain of edges. The information in $H_{0}(K(t))$ gives us useful information on the function $f$ , but being able to assess how connected components (i.e. elements of $H_{0}(K(t))$ ) appear and merge as we vary $t$ would be better. We can do so by introducing the notion of persistent homology.

Refer to caption — Figure 1. The sublevel set filtrations $K(t)$ of a sample of 100 points from a $8$ -dependent stationary Gaussian process along with its $0^{th}$ persistence diagram $PD_{0}$ (upper right).

Given the inclusion maps $\iota_{s,t}\mathrel{\mathop{\mathchar 58\relax}}K(s)\to K(t)$ , for $s\leq t$ there exist linear maps between all homology groups

f^{s,t}_{0}\mathrel{\mathop{\mathchar 58\relax}}H_{0}(K(s))\to H_{0}(K(t)),

which are induced by $\iota_{s,t}$ . The persistent homology groups of the filtration $(K(t))_{t\in\mathbb{R}}$ are the quotient vector spaces

H^{s,t}_{0}(K)\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{im}\,f^{s,t}_{0}\cong Z_{0}(K(s))/\big{(}B_{0}(K(t))\cap Z_{0}(K(s))\big{)},

whose elements represent the cycles that are “born” in $K(s)$ or before and that “die” after $K(s)$ . The dimensions of these vector spaces are the persistent Betti numbers $\beta_{0}^{s,t}$ . Heuristically, a connected component $\gamma\in H_{0}(K(s))$ is born at $K(s)$ if it appears for the first time in $H_{0}(K(s))$ —formally, $\gamma\not\in H_{0}(K(r))$ , for $r<s$ . The component $\gamma\in H_{0}(K(s))$ dies entering $K(t)$ if it merges with an older class (born before $s$ ) entering $H_{k}(K(t))$ . The $0^{th}$ persistent homology of $\mathcal{X}$ , denoted $PH_{0}$ , is the collection of homology groups $H_{0}(K(t))$ and maps $f^{s,t}_{0}$ , for $-\infty<s\leq t\leq\infty$ . All of the information in the persistent homology groups is contained in a multiset in $\mathbb{R}^{2}$ called the persistence diagram (Edelsbrunner and Harer, 2010). The $0^{th}$ persistence diagram of $(K(t))_{t\in\mathbb{R}}$ , denoted $PD_{0}$ , consists of the points $(b,d)$ with multiplicity equal to the number of the classes that are born at $K(b)$ and die entering $K(d)$ . Often, the diagonal $y=x$ is added to this diagram, but we need not consider this here. Formally, we have

PD_{0}=\big{\{}(b,d)\mathrel{\mathop{\mathchar 58\relax}}\text{there exists }\gamma\in PH_{0}\text{ born at }b\text{ that dies entering }d\big{\}},

where $PD_{0}$ is a multiset. Each point $(b,d)$ in $PD_{0}$ can also be represented as a barcode, or interval $[b,d)\subset\mathbb{R}$ (cf. Carlsson and Vejdemo-Johansson, 2021). As such, we may represent $PD_{0}$ as a measure

\xi_{0}=\sum_{(b,d)\in PD_{0}}\delta_{(b,d)},

on $\Delta\mathrel{\mathop{\mathchar 58\relax}}=\{(x,y)\in\bar{\mathbb{R}}^{2}\mathrel{\mathop{\mathchar 58\relax}}-\infty<x<y\leq\infty\}$ . See Figure 1 for an illustration of a persistence diagram associated to a sublevel set filtration of a given stochastic process.

2.3. Probability and persistence

Throughout the paper, let us fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ . For random variables $X,X_{1},X_{2},\dots$ we write $X_{n}\Rightarrow X$ to convey that $X_{n}$ converges weakly to $X$ , i.e. $\mathbb{E}[f(X_{n})]\to\mathbb{E}[f(X)]$ for all bounded, continuous $f$ . We write $X_{n}\overset{P}{\to}X$ to convey that $X_{n}$ converges in probability to $X$ . We say an event $A\in\mathcal{F}$ occurs “a.s.” (almost surely), if $\mathbb{P}(A)=1$ . We use the term stationary throughout this work to refer to the strict stationarity of invariance of finite-dimensional distributions under shifts. A stationary sequence $X_{1},X_{2},\dots$ of random variables is said to be ergodic if any a.s. shift-invariant event $E$ satisfies either $\mathbb{P}(E)=0$ or $\mathbb{P}(E)=1$ .

As we are interested in studying the stochastic behavior of persistence diagrams, we want to associate to each vertex $v_{i}$ a random variable $X_{i}$ for each $i=0,1,2,\dots$ . Consider a stationary sequence of random variables $X_{1},X_{2},\dots$ and define $X_{0}\equiv\infty$ . We then define for $t\in\mathbb{R}$ the filtration

K_{n}(t)\mathrel{\mathop{\mathchar 58\relax}}=\big{\{}\sigma\in K\mathrel{\mathop{\mathchar 58\relax}}\max_{v_{i}\in\sigma}X_{i,n}\leq t\big{\}},

where $X_{0,n}=X_{k,n}=\infty$ for $k>n$ and $X_{k,n}=X_{k}$ otherwise. Furthermore, set $K_{n}=\big{(}K_{n}(t)\big{)}_{t\in\mathbb{R}}$ . Crucially, we can show that

(1)

\beta_{0,n}^{s,t}=\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq t,\bigwedge_{k=j}^{j+i-1}X_{k,n}\leq s\bigg{\}}\mathbf{1}\big{\{}X_{j-1,n}\wedge X_{j+i,n}>t\big{\}}.

We now formalize (1) into a proposition and present a proof.

Proposition 2.1.

The formula (1) holds.

Proof.

Take two vertices $v_{i},v_{j}\in Z_{0}(K_{n}(s))$ . These vertices are equivalent if and only if

v_{i}+v_{j}\in B_{1}(K_{n}(t)),

i.e. if they can be connected by edges lying in $K_{n}(t)$ . Hence, $v_{i}$ and $v_{j}$ must lie in the same connected component in $K_{n}(t)$ . Thus there is a one-to-one correspondence between the number of connected components in $K_{n}(t)$ (which contain a vertex from $K_{n}(s)$ ) and the number of equivalence classes present in $H^{s,t}_{0}(K_{n})$ . Hence, these same classes form a spanning set. Let $[c]$ denote the equivalence class of a chain $c$ . Now take the vertices $[v_{i_{1}}],\dots,[v_{i_{\ell}}]$ that constitute $H^{s,t}_{0}(K_{n})$ (note that $\ell\leq n+1$ ). Then,

a_{i_{1}}[v_{i_{1}}]+\cdots+a_{i_{\ell}}[v_{i_{\ell}}]=[a_{i_{1}}v_{i_{1}}+\cdots+a_{i_{\ell}}v_{i_{\ell}}]=0

if and only if

a_{i_{1}}v_{i_{1}}+\cdots+a_{i_{\ell}}v_{i_{\ell}}\in B_{1}(K_{n}(t)),

where the $a$ terms lie in $\mathbb{Z}_{2}$ . Suppose without loss of generality that

{i_{1}}<\cdots<{i_{\ell}}

As $v_{i_{1}}$ lies in a different connected component from the rest of the vertices, any $1$ -chain of edges in $K_{n}(t)$ including an edge that $v_{i_{1}}$ is a part of, must have a boundary containing a point not equal to $v_{i_{1}}$ and also not equal to $v_{i_{2}},\dots,v_{i_{\ell}}$ . Hence $a_{i_{1}}=0$ , and induction furnishes the other cases. Hence, (1) holds. ∎

Having brought forth the representation of persistent Betti numbers that will prove crucial to the results herein, we turn our attention to persistence diagrams. Let $\xi_{0,n}$ be the measure on $\Delta$ associated to the $0^{th}$ persistence diagram $PD_{0}$ of the filtration $K_{n}=\big{(}K_{n}(t)\big{)}_{t\in\mathbb{R}}$ . Note that

\beta_{0,n}^{s,t}=\xi_{0,n}\big{(}(-\infty,s]\times(t,\infty]\big{)}.

If we let

R=(s_{1},s_{2}]\times(t_{1},t_{2}],

for $-\infty<s_{1}<s_{2}\leq t_{1}<t_{2}\leq\infty$ , then

(2)

\xi_{0,n}(R)=\beta_{0,n}^{s_{2},t_{1}}-\beta_{0,n}^{s_{2},t_{2}}-\beta_{0,n}^{s_{1},t_{1}}+\beta_{0,n}^{s_{1},t_{2}},

due to the so-called “Fundamental Lemma of Persistent Homology” (Edelsbrunner and Harer, 2010). If $R$ has the above representation, we will say that $s_{1},s_{2},t_{1},t_{2}$ are the coordinates of $R$ . We define the class $\mathcal{R}$ of sets by

\mathcal{R}\mathrel{\mathop{\mathchar 58\relax}}=\big{\{}(s_{1},s_{2}]\times(t_{1},t_{2}]\mathrel{\mathop{\mathchar 58\relax}}-\infty<s_{1}<s_{2}\leq t_{1}<t_{2}\leq\infty\big{\}}.

An important result holds for the class $\mathcal{R}$ .

Lemma 2.2.

$\mathcal{R}$ is a convergence-determining class for weak convergence on $\Delta$ equipped with the Borel $\sigma$ -algebra, $\mathcal{B}(\Delta)$ . Namely, if $(\mu_{n})_{n}$ and $\mu$ are probability measures on $\Delta$ and

\mu_{n}(R)\to\mu(R),\quad n\to\infty,

for all $R\in\mathcal{R}$ such that $\mu(\partial R)=0$ , then

\mu_{n}\Rightarrow\mu,\quad n\to\infty.

Furthermore, for each probability measure $\mu$ on $\Delta$ there is a countable convergence-determining class $\mathcal{R}_{\mu}\subset\mathcal{R}$ for $\mu$ .

Proof.

We will adapt the proof of Theorem A.2 from Hiraoka et al. (2018). First, it is clear that $\mathcal{R}$ is closed under finite intersections, so we have satisfied the first condition of Theorem 2.4 in Billingsley (1999) (i.e. that $\mathcal{R}$ is a $\pi$ -system). It is also evident that $\Delta$ is separable. Now, for any $z\in\Delta$ if we denote

\mathcal{R}_{z,\epsilon}\mathrel{\mathop{\mathchar 58\relax}}=\{R\in\mathcal{R}\mathrel{\mathop{\mathchar 58\relax}}z\in R^{\circ}\subset R\subset B(z,\epsilon)\},

then the class of boundaries $\partial\mathcal{R}_{z,\epsilon}$ contains uncountably many disjoint sets, regardless of if $z=(s,\infty)$ or $(s,t)$ , where $t<\infty$ (in the former case $R^{\circ}=(s_{1},s_{2})\times(t_{1},\infty]$ ). Thus $\mathcal{R}$ is a convergence-determining class by Theorem 2.4 of Billingsley (1999).

For the final part of the proof, let us fix a probability measure $\mu$ and choose an open set $U\subset\Delta$ . Note that for every $z\in U$ , there is an $\epsilon>0$ such that $B(z,\epsilon)\subset U$ . By the first part of this proof, for each of these $B(z,\epsilon)$ there exists a set $R_{z}\equiv R^{U}_{z}\in\mathcal{R}_{z,\epsilon}$ such that $\mu(\partial R_{z})=0$ and hence we have

U=\bigcup_{z\in U}R_{z}=\bigcup_{z\in U}R^{\circ}_{z},

and $U$ is the union of sets with $\mu$ -null boundaries. By $\Delta$ separable, there exists a countable subcover $\{R^{U}_{z_{i}}\}_{i=1}^{\infty}$ of $U$ . Also, there exists a countable basis $\{U_{j}\}_{j=1}^{\infty}$ of $\Delta$ . Hence, if we denote $R_{i,j}\mathrel{\mathop{\mathchar 58\relax}}=R^{U_{j}}_{z_{i}}$ then

U_{j}=\bigcup_{i=1}^{\infty}R_{i,j}=\bigcup_{i=1}^{\infty}R^{\circ}_{i,j}.

If we let $\mathcal{R}_{\mu}$ be the class of finite intersections of the sets $R_{i,j}$ . As the boundary of an intersection is a subset of the union of the boundaries, each element of $\mathcal{R}_{\mu}$ has a $\mu$ -null boundary. Furthermore, every open set in $\Delta$ is the countable union of elements of $\mathcal{R}_{\mu}$ . Hence, we apply Theorem 2.2 in Billingsley (1999) and the result holds.

∎

An important result holds for the measure $\xi_{0,n}$ . Namely that the value $\xi_{0,n}(\Delta)$ is equal to the number of local minima of $X_{0,n},X_{1,n},\dots,X_{n,n},X_{n+1,n}$ .

Proposition 2.3.

Suppose that $X_{1},X_{2},\dots$ is a stationary sequence of random variables with $\mathbb{P}(X_{1}=X_{2})=0$ . Then

\xi_{0,n}(\Delta)=\sum_{i=1}^{n}\mathbf{1}\big{\{}X_{i,n}<X_{i-1,n}\wedge X_{i+1,n}\big{\}}

Proof.

The case when $n=1$ is trivial, so suppose that $n\geq 2$ . As the underlying stochastic process is stationary and $\mathbb{P}(X_{1}=X_{2})=0$ then every value $X_{1},X_{2},\dots$ is distinct with probability 1. Let $a_{i}\equiv X_{(i),n}$ be the order statistics of $X_{1,n},\dots,X_{n,n}$ —which are distinct with probability 1—and let $v_{(i)}$ be the associated vertices (see above). If we define

K_{i}\mathrel{\mathop{\mathchar 58\relax}}=K_{n}(a_{i}),\quad i=1,\dots,n,

with $K_{0}=\emptyset$ , then $K_{0}\subset K_{1}\subset\cdots\subset K_{n}$ and $K_{i+1}$ contains all the simplices of $K_{i}$ along with the 0-simplex $v_{(i+1)}$ and any edges containing it. If $m>\ell$ then there are $\alpha$ points at $(a_{\ell},a_{m})\in\xi_{0,n}$ if and only if $\xi_{0,n}((a_{\ell-1},a_{\ell}]\times(a_{m-1},a_{m}])=\alpha$ —see p. 152 in Edelsbrunner and Harer (2010). By Proposition 2.1, we have that

	$\displaystyle\xi_{0,n}((a_{\ell-1},a_{\ell}]\times(a_{m-1},a_{m}])$
	$\displaystyle\ =\beta_{0,n}^{a_{\ell},a_{m-1}}-\beta_{0,n}^{a_{\ell},a_{m}}-\beta_{0,n}^{a_{\ell-1},a_{m-1}}+\beta_{0,n}^{a_{\ell-1},a_{m}}$
	$\displaystyle\ =\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}\mathbf{1}\bigg{\{}\bigwedge_{k=j}^{j+i-1}X_{k,n}=a_{\ell}\bigg{\}}$
	$\displaystyle\ \times\Bigg{[}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq a_{m-1},X_{j-1,n}\wedge X_{j+i,n}>a_{m-1}\bigg{\}}$
	$\displaystyle\phantom{\ \times\Bigg{[}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}}-\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq a_{m},X_{j-1,n}\wedge X_{j+i,n}>a_{m}\bigg{\}}\Bigg{]}.$

Now, $\xi_{0,n}(\Delta)=\sum_{\ell=1}^{n-1}\sum_{m=\ell+1}^{n}\xi_{0,n}((a_{\ell-1},a_{\ell}]\times(a_{m-1},a_{m}])$ so by cancelling sums—and the fact that $n\geq 2$ implies that $X_{j-1,n}\wedge X_{j+i,n}>a_{n}$ cannot happen—we have that

	$\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}\sum_{\ell=1}^{n-1}\mathbf{1}\bigg{\{}\bigwedge_{k=j}^{j+i-1}X_{k,n}=a_{\ell}\bigg{\}}$
	$\displaystyle\phantom{\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}\sum_{\ell=1}^{n-1}}\qquad\times\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq a_{\ell},X_{j-1,n}\wedge X_{j+i,n}>a_{\ell}\bigg{\}}$
(3)		$\displaystyle=\sum_{j=1}^{n}\sum_{\ell=1}^{n-1}\mathbf{1}\big{\{}X_{j,n}=a_{\ell},X_{j-1,n}\wedge X_{j+1,n}>a_{\ell}\},$

because the only way the maximum and minimum of a collection of $i$ of random variables are idenitical is if they’re constant—which is only possible if $i=1$ as the $X_{i,n},i=1,\dots,n$ are almost surely distinct. The desired formula follows from applying this same uniqueness to (3).

∎

To finish this section, we must introduce the restricted measure on the set $\tilde{\Delta}\mathrel{\mathop{\mathchar 58\relax}}=\Delta\cap\mathbb{R}^{2}$ —equipped with the usual Borel sub $\sigma$ -algebra $\mathcal{B}(\tilde{\Delta})$ —defined by

\tilde{\xi}_{0,n}(A)\mathrel{\mathop{\mathchar 58\relax}}=\xi_{0,n}(A),\quad A\in\mathcal{B}(\tilde{\Delta}).

Note that as $\Delta\cap\mathbb{R}^{2}$ is Borel subset of $\Delta$ that $\mathcal{B}(\tilde{\Delta})\subset\mathcal{B}(\Delta)$ . To reduce notational clutter, we will mostly write $\tilde{\xi}_{0,n}(\Delta)$ in place of $\tilde{\xi}_{0,n}(\tilde{\Delta})$ from here on out, unless otherwise noted.

3. Strong law of large numbers

In this section we establish our strong law of large numbers for sublevel set persistence diagrams for a very broad class of sets and functions. We do this for the class of bounded, continuous functions initially via a weak convergence argument, and proceed to extend our result to a class of unbounded functions which are of great practical use in topological data analysis. Along the way, we give an explicit representation for the limiting persistent Betti number for i.i.d. sequences.

Theorem 3.1.

Consider a stationary and ergodic sequence $\mathcal{X}=(X_{1},X_{2},\dots)$ where each $X_{i}$ has distribution $F$ and density $f$ such that $\mathbb{P}(X_{1}=X_{2})=0$ . For the random probability measure $\xi_{0,n}/\xi_{0,n}(\Delta)$ induced by $\mathcal{X}$ there exists a probability measure $\xi_{0}$ on $\Delta$ such that

\frac{\xi_{0,n}}{\xi_{0,n}(\Delta)}\Rightarrow\xi_{0}\ \ \mathrm{a.s.},\quad n\to\infty.

Additionally, if we define $\tilde{\xi}_{0}\equiv\xi_{0}$ on $\mathcal{B}(\tilde{\Delta})$ then

\frac{\tilde{\xi}_{0,n}}{\tilde{\xi}_{0,n}(\Delta)}\Rightarrow\tilde{\xi}_{0}\ \ \mathrm{a.s.},\quad n\to\infty.

Proof.

We will begin by establishing the almost sure convergence of the persistent Betti numbers $\beta_{0,n}^{s,t}/n$ for $-\infty<s\leq t\leq\infty$ . Recall that

	$\displaystyle\frac{\beta_{0,n}^{s,t}}{n}$	$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq t,\bigwedge_{k=j}^{j+i-1}X_{k,n}\leq s\bigg{\}}\mathbf{1}\big{\{}X_{j-1,n}\wedge X_{j+i,n}>t\big{\}}$
		$\displaystyle=\frac{1}{n}\sum_{j=1}^{n}\sum_{i=1}^{n-j+1}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq t,\bigwedge_{k=j}^{j+i-1}X_{k,n}\leq s\bigg{\}}\mathbf{1}\big{\{}X_{j-1,n}\wedge X_{j+i,n}>t\big{\}}$

Define for $m\in\mathbb{N}\cup\{\infty\}$ the indicator random variable

(4)

Y_{j,n}^{m}(s,t)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{m}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq t,\bigwedge_{k=j}^{j+i-1}X_{k,n}\leq s\bigg{\}}\mathbf{1}\big{\{}X_{j-1,n}\wedge X_{j+i,n}>t\big{\}},

with the indicators $Y_{j}^{m}(s,t)$ defined as $Y^{m}_{j,n}(s,t)$ with the second subscript $n$ dropped. If we fix $m$ , we have for $n\geq m$ that

\beta_{0,n}^{s,t}=\sum_{j=1}^{n}Y_{j,n}^{n-j+1}(s,t)\geq\sum_{j=1}^{n-m+1}Y_{j,n}^{n-j+1}(s,t)\geq\sum_{j=1}^{n-m+1}Y_{j}^{m}(s,t),

which yields

\beta_{0,n}^{s,t}\geq\sum_{j=2}^{n+1}Y_{j}^{m}(s,t)-(m+1).

Similarly, we see that

\beta_{0,n}^{s,t}\leq 1+\sum_{j=1}^{n}Y_{j,n}^{n-j}(s,t)\leq 2+\sum_{j=2}^{n+1}Y_{j}^{\infty}(s,t),

because

\sum_{j=1}^{n}\mathbf{1}\bigg{\{}\bigvee_{k=j}^{n}X_{k,n}\leq t,\bigwedge_{k=j}^{n}X_{k,n}\leq s\bigg{\}}\mathbf{1}\big{\{}X_{j-1,n}>t\big{\}}\in\{0,1\}.

It is readily observed for fixed $t\geq s$ that $Y_{2}^{m}(s,t),Y_{3}^{m}(s,t),\dots$ are indicator random variables and form a stationary and ergodic sequence, owing to Theorem 7.1.3 in Durrett (2010), for example. Thus, Birkhoff’s ergodic theorem implies that for any $m\in\mathbb{N}$ we have

\mathbb{E}[Y_{2}^{m}(s,t)]\leq\liminf_{n\to\infty}\frac{\beta_{0,n}^{s,t}}{n}\leq\limsup_{n\to\infty}\frac{\beta_{0,n}^{s,t}}{n}\leq\mathbb{E}[Y^{\infty}_{2}(s,t)],\quad\mathrm{a.s.}

The monotone convergence theorem then implies that

n^{-1}\beta_{0,n}^{s,t}\overset{\text{a.s.}}{\to}\mathbb{E}[Y_{2}^{\infty}(s,t)],\quad n\to\infty.

To establish the convergence of $\xi_{0,n}(\Delta)/n$ , it suffices to recall that from Proposition 2.3 the total number of points in the persistence diagram $\xi_{0,n}(\Delta)$ is equal to the number of local minima of $\mathcal{X}$ . Therefore, the ergodic theorem once again implies that $\xi_{0,n}(\Delta)/n$ converges a.s. to $\mathbb{P}(X_{2}<X_{1}\wedge X_{3})$ and

(5)

\frac{\xi_{0,n}\big{(}(-\infty,s]\times(t,\infty]\big{)}}{\xi_{0,n}(\Delta)}\to\frac{\mathbb{E}[Y_{2}^{\infty}(s,t)]}{\mathbb{P}(X_{2}<X_{1}\wedge X_{3})},\ \ \mathrm{a.s.},\quad n\to\infty.

(By our assumptions we must have that $P(X_{2}<X_{1}\wedge X_{3})>0$ ). Define a set function $\bar{\xi}_{0}$ by

\bar{\xi}_{0}\big{(}(-\infty,s]\times(t,\infty])\big{)}\mathrel{\mathop{\mathchar 58\relax}}=\frac{\mathbb{E}[Y_{2}^{\infty}(s,t)]}{\mathbb{P}(X_{2}<X_{1}\wedge X_{3})}

which can likewise be defined on $\mathcal{R}$ in a straightforward manner, by (2). It is clear that the convergence in (5) holds for any set in $\mathcal{R}$ as well. As $\mathcal{R}$ is a semiring which generates the Borel $\sigma$ -algebra $\mathcal{B}(\Delta)$ on $\Delta$ (as $\Delta$ is separable), then $\bar{\xi}_{0}$ extends uniquely to a probability measure $\xi_{0}$ on $\mathcal{B}(\Delta)$ , provided that $\bar{\xi}_{0}$ is countably additive on $\mathcal{R}$ . By Lemma 2.2, there is a countable convergence-determining class $\mathcal{R}_{0}$ for $\xi_{0}$ . We have shown thus far that

\mathbb{P}\Bigg{(}\lim_{n\to\infty}\frac{\xi_{0,n}(R)}{\xi_{0,n}(\Delta)}\to\xi_{0}(R),\text{ for any }R\in\mathcal{R}_{0}\Bigg{)}=1,

so convergence for all sets in $\mathcal{B}(\Delta)$ with $\xi_{0}$ -null boundary follows (with probability 1). It remains to demonstrate that $\bar{\xi}_{0}$ is countably additive on $\mathcal{R}$ . Let

(s_{1},s_{2}]\times(t_{1},t_{2}]=\bigcup_{i=1}^{\infty}(s_{1,i},s_{2,i}]\times(t_{1,i},t_{2,i}],

where $(s_{1,i},s_{2,i}]\times(t_{1,i},t_{2,i}]$ are disjoint. Then, almost surely,

	$\displaystyle\bar{\xi}_{0}\big{(}(s_{1},s_{2}]\times(t_{1},t_{2}]\big{)}$	$\displaystyle=\lim_{n\to\infty}\frac{\xi_{0,n}\big{(}(s_{1},s_{2}]\times(t_{1},t_{2}]\big{)}}{\xi_{0,n}(\Delta)}$
		$\displaystyle=\lim_{n\to\infty}\sum_{i=1}^{\infty}\frac{\xi_{0,n}\big{(}(s_{1,i},s_{2,i}]\times(t_{1,i},t_{2,i}]\big{)}}{\xi_{0,n}(\Delta)}$
		$\displaystyle=\sum_{i=1}^{\infty}\bar{\xi}_{0}\big{(}(s_{1,i},s_{2,i}]\times(t_{1,i},t_{2,i}]\big{)},$

by the monotone convergence theorem.

To finish the proof, note that it is the case²²2This fact implies that $\xi_{0}$ is supported on $\tilde{\Delta}$ . that $\tilde{\xi}_{0,n}(\tilde{\Delta})\sim\xi_{0,n}(\Delta)$ —as they both tend to infinity and differ by 1. Also, we have that for any set $A\in\mathcal{B}(\tilde{\Delta})$ —which is also a Borel subset of $\Delta$ —if $\xi_{0}(\partial A)=0$ , then almost surely

\frac{\tilde{\xi}_{0,n}(A)}{\tilde{\xi}_{0,n}(\Delta)}\sim\frac{\xi_{0,n}(A)}{\xi_{0,n}(\Delta)}\to\xi_{0}(A),\quad n\to\infty.

As $\xi_{0}(A)=\tilde{\xi_{0}}(A)$ for $A\in\mathcal{B}(\tilde{\Delta})$ , the proof is finished. ∎

Remark 3.2.

In Theorem 3.1 we assumed that $\mathbb{P}(X_{1}=X_{2})=0$ in our stationary sequence, to ensure consecutive points are distinct, as stated in Proposition 2.3. It seems straightforward to generalize this result to the situation where consecutive points can be identical, by accounting for this in the proof of Proposition 2.3, and ensuring that the number of points in $\xi_{0,n}$ tends to infinity.

Before seeing an example of the strong law in action, we will establish a result that will provide us an explicit representation of the limiting measure. Let us define the quantity

p_{i}(s,t)\mathrel{\mathop{\mathchar 58\relax}}=\mathbb{P}\Big{(}\bigcup_{k=1}^{i}\big{\{}X_{1}\leq t,\dots,X_{k}\leq s,\dots,X_{i}\leq t\}\Big{)},

which represents the probability that there is some index $k$ such that $X_{k}\leq s$ and all other random variables are less than or equal to $t$ . In the setup with $X_{i}$ all i.i.d. with distribution function $F$ we have

p_{i}(s,t)=F(t)^{i}-\big{(}F(t)-F(s)\big{)}^{i}

and

	$\displaystyle\mathbb{E}[\beta_{0,n}^{s,t}]$	$\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}\mathbb{P}\bigg{(}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq t,\bigwedge_{k=j}^{j+i-1}X_{k,n}\leq s,\text{ and }X_{j-1,n}\wedge X_{j+i,n}>t\bigg{)}$
(6)			$\displaystyle=p_{n}(s,t)+2p_{n-1}(s,t)(1-F(t))+\sum_{i=1}^{n-2}\bigg{(}2p_{i}(s,t)(1-F(t))+(n-i-1)p_{i}(s,t)(1-F(t))^{2}\bigg{)}.$

We will assume that $0<F(s)<1$ , as if $F(s)=0$ then $\beta^{s,t}_{0,n}\equiv 0$ and if $F(s)=1$ then $\beta^{s,t}_{0,n}\equiv 1$ . Dividing (6) by $n$ we can see that

	$\displaystyle\frac{\mathbb{E}[\beta_{0,n}^{s,t}]}{n}$	$\displaystyle\sim\frac{(1-F(t))^{2}}{n}\sum_{i=1}^{n}(n-i+1)p_{i}(s,t)$
		$\displaystyle=\frac{(1-F(t))^{2}}{n}\sum_{i=1}^{n}(n-i+1)[F(t)^{i}-\big{(}F(t)-F(s)\big{)}^{i}]$

as the other terms are finite or tend to zero upon dividing by $n$ . Let us make the substitution $i=n-j+1$ and consider a general $a\in(0,1]$ with $b=a^{-1}$ . Thus,

	$\displaystyle\sum_{i=1}^{n}(n-i+1)a^{i}$	$\displaystyle=a^{n}\sum_{j=1}^{n}jb^{j-1}$
		$\displaystyle=a^{n}\Bigg{[}\frac{nb^{n+1}-(n+1)b^{n}+1}{(b-1)^{2}}\Bigg{]}$
(7)			$\displaystyle=\frac{nb-(n+1)+a^{n}}{(b-1)^{2}}$

by differentiating $\sum_{i=1}^{n}x^{i}=(x^{n+1}-x)/(x-1)$ with respect to $x$ . We have the following pleasing result for the limiting expectation for the persistent Betti number in this simplified i.i.d case.

Proposition 3.3.

For $X_{i}$ i.i.d. having distribution $F$ , we have that

\frac{\mathbb{E}[\beta_{0,n}^{s,t}]}{n}\to\frac{(1-F(t))F(s)}{1-F(t)+F(s)},

for any $-\infty<s\leq t\leq\infty$ with $F(s)\in(0,1)$ and $0$ otherwise.

Proof.

Dividing by $n$ and taking the limit in (7) for the two cases $a=F(t)$ and $a=F(t)-F(s)$ gives

\frac{(1-F(t))^{2}}{1/F(t)-1}=(1-F(t))F(t),

and

\frac{(1-F(t))^{2}}{1/[F(t)-F(s)]-1}=\frac{(1-F(t))^{2}[F(t)-F(s)]}{1-F(t)+F(s)}.

Simplifying the above two expressions yields the ultimate result. ∎

Example 3.4.

If the stationary and ergodic sequence in Theorem 3.1 is i.i.d, Proposition 3.3 shows we can characterize the limiting probability measure $\xi_{0}$ quite nicely. We note that

\xi_{0}\big{(}(-\infty,s]\times(t,\infty]\big{)}=\frac{3(1-F(t))F(s)}{1-F(t)+F(s)}

for all $\infty<s\leq t\leq\infty$ as $\mathbb{P}(X_{2}<X_{1}\wedge X_{3})=1/3$ . Therefore, $\xi_{0}$ admits a probability density

	$\displaystyle-\frac{\partial^{2}}{\partial x\partial y}\Bigg{[}\frac{3(1-F(y))F(x)}{1-F(y)+F(x)}\Bigg{]}$
	$\displaystyle\phantom{-\frac{\partial^{2}}{\partial x\partial y}}=\frac{6f(x)f(y)(1-F(y))F(x)}{(1-F(y)+F(x))^{3}}.$

This density facilities the simulation of random variables according to the limiting persistence distribution $\xi^{\text{NULL}}_{0}$ in the case that $\mathcal{X}$ corresponds to i.i.d. noise. After a Monte Carlo random sample is generated from this distribution, we may test for “significant” points $(b,d)$ in the diagram $\xi_{0,n}$ , based off of what we would expect from $\xi^{\text{NULL}}_{0}$ .

Of particular importance to us is the partial derivative

	$\displaystyle\frac{\partial}{\partial x}\Bigg{[}\frac{3(1-F(y))F(x)}{1-F(y)+F(x)}\Bigg{]}$
(8)		$\displaystyle\phantom{-\frac{\partial^{2}}{\partial x\partial y}}=\frac{3f(x)(1-F(y))^{2}}{(1-F(y)+F(x))^{2}}.$

If we set $y=x+\ell$ , then (8) evaluates to

\displaystyle 3\Bigg{(}\frac{1-F(x+\ell)}{1-F(x+\ell)+F(x)}\Bigg{)}^{2}f(x)

Define $\Delta_{\ell}\mathrel{\mathop{\mathchar 58\relax}}=\{(x,y)\in\Delta\mathrel{\mathop{\mathchar 58\relax}}y-x>\ell\}$ for $\ell\geq 0$ . As a result of the above discussion, we have the following corollary.

Corollary 3.5.

For $X_{1},X_{2},\dots$ i.i.d. with distribution function $F$ satisfying the conditions of Theorem 3.1, we have that

\xi_{0}(\Delta_{\ell})=3\mathbb{E}\Bigg{[}\frac{1-F(X+\ell)}{1-F(X+\ell)+F(X)}\Bigg{]}^{2}

where $X\overset{d}{=}X_{1}$ .

Example 3.6.

Corollary 3.5 implies that for $F(t)$ uniform on $[0,1]$ we have for $0<\ell<1$ that

	$\displaystyle\xi_{0}(\Delta_{\ell})$	$\displaystyle=3\int_{0}^{1-\ell}\bigg{(}\frac{1-\ell-x}{1-\ell}\bigg{)}^{2}\operatorname{d\!}{x}.$
		$\displaystyle=1-\ell,$

This is a rather interesting, given that there is no a priori reason that uniform noise should also produce asymptotically uniformly distributed persistence lifetimes.

Before addressing strong laws for unbounded functions, we conclude with a corollary of Theorem 3.1, establishing a Glivenko-Cantelli result for persistence lifetimes. We omit the proof of Corollary 3.7 as it is proved in exactly the same manner as the Glivenko-Cantelli theorem—see Theorem 1.3 in Dudley (2014).

Corollary 3.7.

Suppose the conditions on the sequence $\mathcal{X}$ stated in Theorem 3.1 hold. Then we have

\sup_{\ell\in[0,\infty)}\Bigg{|}\frac{\xi_{0,n}(\Delta_{\ell})}{\xi_{0,n}(\Delta)}-\xi_{0}(\Delta_{\ell})\Bigg{|}\to 0\ \mathrm{a.s.},\quad n\to\infty.

3.1. SLLN for unbounded functions

At this point, we have established almost surely that

\tilde{\xi}_{0,n}(f)/\tilde{\xi}_{0,n}(\Delta)\to\tilde{\xi}_{0}(f),

for any bounded, continuous real-valued function $f$ on $\tilde{\Delta}$ , when $\tilde{\xi}_{0,n}$ is induced by a stationary and ergodic sequence of random variables (similar for $\xi_{0,n}$ ). In general, if $f$ is continuous, nonnegative function and $f\wedge M$ is the function that equals $M$ when $f\geq M$ , then almost surely

\tilde{\xi}_{0,n}(f\wedge M)/\tilde{\xi}_{0,n}(\Delta)=\frac{\sum_{(b,d)\in\tilde{\xi}_{0,n}}f(b,d)\wedge M}{\sum_{(b,d)\in\tilde{\xi}_{0,n}}1}\to\int_{\tilde{\Delta}}f(x,y)\wedge M\,\tilde{\xi}_{0}(\operatorname{d\!}{x},\operatorname{d\!}{y}),\quad n\to\infty,

for all $M>0$ . Following this line of inquiry, we establish a result which yields convergence results for a large class of persistence statistics often seen in practice, including many of the functions for which convergence holds for geometric complexes in Divol and Polonik (2019), though we make no requirements on the behavior near the diagonal nor do we require polynomial growth. Prior to stating the result, it is necessary to define the notion of largely nondecreasing. We say that an unbounded function $g\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}_{+}\to\mathbb{R}_{+}$ is largely nondecreasing if there exists an $M>0$ such that $\{x\mathrel{\mathop{\mathchar 58\relax}}g(x)\geq M\}$ is non-empty and $g$ is nondecreasing on $[g^{\leftarrow}(M),\infty)$ where $g^{\leftarrow}(M)=\inf\{x\mathrel{\mathop{\mathchar 58\relax}}g(x)\geq M\}$ . Furthermore, recall that the function $g$ is coercive if $g(x)\to\infty$ as $x\to\infty$ .

Theorem 3.8.

Assume the conditions of Theorem 3.1 and suppose that $f(b,d)=g(d-b)$ and $g\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a continuous, coercive, and largely nondecreasing function with $\mathbb{E}\big{[}g(2|X_{1}|)^{1+\epsilon}\big{]}<\infty$ for some $\epsilon>0$ . If $\tilde{\xi}_{0}(f)<\infty$ , then

\tilde{\xi}_{0,n}(f)/\tilde{\xi}_{0,n}(\Delta)\to\tilde{\xi}_{0}(f),\ \ \mathrm{a.s.},\quad n\to\infty.

Proof.

Before beginning, fix any $M>0$ such that $g$ is nondecreasing on $[g^{\leftarrow}(M),\infty)$ . We will focus our proof on the case where the marginal distribution $F$ can take negative and positive values, but the proofs follow from a simplified version of the argument below when the support of $F$ is restricted to a half-line. To show that

\tilde{\xi}_{0,n}(f)/\tilde{\xi}_{0,n}(\Delta)\to\tilde{\xi}_{0}(f),

for $f$ as in the statement of the theorem, it will suffice to first bound the quantity

	$\displaystyle\frac{\tilde{\xi}_{0,n}(f)}{\tilde{\xi}_{0,n}(\Delta)}-\frac{\tilde{\xi}_{0,n}(f\wedge M)}{\tilde{\xi}_{0,n}(\Delta)}$	$\displaystyle=\frac{\tilde{\xi}_{0,n}\big{(}(f-M)_{+}\big{)}}{\tilde{\xi}_{0,n}(\Delta)}$
(9)			$\displaystyle=\tilde{\xi}_{0,n}(\Delta)^{-1}\sum_{\begin{subarray}{c}(b,d)\in\xi_{0,n},\\ f(b,d)\geq M\end{subarray}}f(b,d).$

Recall that $f(b,d)=g(d-b)$ . In this situation, we have that the unnormalized form of (9) equals

	$\displaystyle\sum_{d-b\geq g^{\leftarrow}(M)}g(d-b)$	$\displaystyle=\sum_{\begin{subarray}{c}d-b\geq g^{\leftarrow}(M),\\ b\geq 0\end{subarray}}g(d-b)+\sum_{\begin{subarray}{c}d-b\geq g^{\leftarrow}(M),\\ \ b<0,\,d<0\end{subarray}}g(d-b)+\sum_{\begin{subarray}{c}d-b\geq g^{\leftarrow}(M),\\ \ b<0,\,d\geq 0\end{subarray}}g(d-b)$
		$\displaystyle\leq\sum_{\begin{subarray}{c}d\geq g^{\leftarrow}(M)\end{subarray}}g(d)+\sum_{\begin{subarray}{c}-b\geq g^{\leftarrow}(M)\end{subarray}}g(-b)+\sum_{\begin{subarray}{c}d-b\geq g^{\leftarrow}(M),\\ \ b<0,\,d\geq 0\end{subarray}}g(2d)+g(-2b)$
(10)			$\displaystyle\leq\sum_{\begin{subarray}{c}d\geq g^{\leftarrow}(M)\end{subarray}}g(d)+\sum_{\begin{subarray}{c}-b\geq g^{\leftarrow}(M)\end{subarray}}g(-b)+\sum_{\begin{subarray}{c}2\max\{d,-b\}\geq g^{\leftarrow}(M),\\ \ b<0,\,d\geq 0\end{subarray}}g(2d)+g(-2b),$

because of the fact $g(d-b)\leq g\big{(}2\max\{d,-b\}\big{)}\leq g(2d)+g(-2b)$ when $b<0$ , $d\geq 0$ and we have $d-b\geq g^{\leftarrow}(M)$ . Furthermore,

	$\displaystyle\sum_{\begin{subarray}{c}2\max\{d,-b\}\geq g^{\leftarrow}(M),\\ \ b<0,\,d\geq 0\end{subarray}}g(2d)$	$\displaystyle=\sum_{(b,d)\in\xi_{0,n}}g(2d)\mathbf{1}\big{\{}2\max\{d,-b\}\geq g^{\leftarrow}(M)\big{\}}\big{(}\mathbf{1}\big{\{}d>-b\big{\}}+\mathbf{1}\big{\{}d\leq-b\big{\}}\big{)}$
		$\displaystyle=\sum_{(b,d)\in\xi_{0,n}}g(2d)\mathbf{1}\big{\{}2d\geq g^{\leftarrow}(M)\big{\}}\mathbf{1}\big{\{}d>-b\big{\}}$
		$\displaystyle\qquad\qquad\qquad+\sum_{(b,d)\in\xi_{0,n}}g(2d)\mathbf{1}\big{\{}-2b\geq g^{\leftarrow}(M)\big{\}}\mathbf{1}\big{\{}d\leq-b\big{\}}$
		$\displaystyle\leq\sum_{\begin{subarray}{c}2d\geq g^{\leftarrow}(M)\end{subarray}}g(2d)+\sum_{\begin{subarray}{c}-2b\geq g^{\leftarrow}(M)\end{subarray}}g(-2b).$

This occurs as $g(x)\leq g(y)$ if $y\geq g^{\leftarrow}(M)\vee x$ . With a similar argument for the $g(-2b)$ term, we can see that (10) is bounded above by

\sum_{\begin{subarray}{c}2d\geq g^{\leftarrow}(M)\end{subarray}}3g(2d)+\sum_{\begin{subarray}{c}-2b\geq g^{\leftarrow}(M)\end{subarray}}3g(-2b)

By a similar argument to Proposition 2.3 occurs at $d=X_{i}$ if and only $X_{i}$ is a local maxima. Birkhoff’s ergodic theorem then implies that

\sum_{2d\geq g^{\leftarrow}(M)}g(2d)/n\to\mathbb{E}\big{[}g(2X_{2})\mathbf{1}\big{\{}X_{2}>X_{1}\vee X_{3}\big{\}}\mathbf{1}\big{\{}2X_{2}\geq g^{\leftarrow}(M)\big{\}}\big{]},\quad\text{a.s.},

as $n\to\infty$ . Hölder’s inequality then implies that for $p>1$ and $q=p/(p-1)$ ,

\displaystyle\mathbb{E}\big{[}g(2X_{2})\mathbf{1}\big{\{}X_{2}>X_{1}\vee X_{3}\big{\}}\mathbf{1}\big{\{}2X_{2}\geq g^{\leftarrow}(M)\big{\}}\big{]}

\displaystyle\leq\Bigg{(}\mathbb{E}[g(2|X_{2}|)^{p}]\Bigg{)}^{1/p}\Bigg{(}\mathbb{P}(2|X_{2}|\geq g^{\leftarrow}(M))\Bigg{)}^{1/q}

By assumption, $\mathbb{E}[g(2|X_{2}|)^{p}]<\infty$ for some $p>1$ , so that coercivity of $g$ entails we may choose $M>0$ large enough such that

\mathbb{E}\big{[}g(2X_{2})\mathbf{1}\big{\{}X_{2}>X_{1}\vee X_{3}\big{\}}\mathbf{1}\big{\{}2X_{2}\geq g^{\leftarrow}(M)\big{\}}\big{]}<\epsilon\mathbb{P}(X_{2}<X_{1}\wedge X_{3})/18.

Therefore, for such an $M$ we have

\limsup_{n\to\infty}\sum_{2d\geq g^{\leftarrow}(M)}3g(2d)/\tilde{\xi}_{0,n}(\Delta)<\epsilon/6,\ \ \text{a.s.}

A similar argument holds for the term

\sum_{\begin{subarray}{c}-2b\geq g^{\leftarrow}(M)\end{subarray}}3g(-2b),

so the additivity of $\limsup$ furnishes that

\limsup_{n\to\infty}\sum_{d-b\geq g^{\leftarrow}(M)}g(d-b)/\tilde{\xi}_{0,n}(\Delta)<\epsilon/3,\ \ \text{a.s.}

By Theorem 3.1 and the triangle inequality, it remains to show that

\tilde{\xi}_{0}\big{(}(f-M^{\prime})_{+}\big{)}<\epsilon/3

for some $M^{\prime}\geq M$ , which follows from $\tilde{\xi}_{0}(f)<\infty$ . ∎

If all $X_{i}$ are nonnegative, we have an easy corollary to Theorem 3.8. We omit the proof as it follows directly from the one above.

Corollary 3.9.

If $X_{i}\geq 0$ for all $i=1,2,\dots$ then Theorem 3.8 holds for $f(b,d)=g(d+b)$ .

The utility of Theorem 3.8 can be seen in the following section.

3.2. Strong law of large numbers: two examples

Strong laws of large numbers can be established from Theorem 3.8 for various quantities used in topological data science called persistence statistics. For instance, we have a strong law of large numbers for degree- $p$ total persistence³³3See Cohen-Steiner et al. (2010) for a definition and Divol and Polonik (2019) for the geometric complex result, provided that

\mathbb{E}\big{[}|X_{1}|^{p+\epsilon}\big{]}<\infty.

A more difficult example is persistent entropy (Merelli et al., 2015; Atienza et al., 2020). Persistent entropy has been used as part of a suite of statistics in the studies of Chung et al. (2021, 2022, 2024) and Thomas et al. (2024), as well as to detect activation in the immune system (Rucco et al., 2016), and to detect structure in nanoparticle images (Thomas et al., 2023; Crozier et al., 2024). The definition (excluding the longest barcode) is

E(X_{1},\dots,X_{n})\equiv E_{n}\mathrel{\mathop{\mathchar 58\relax}}=-\sum_{(b,d)\in\tilde{\xi}_{0,n}}\frac{d-b}{L_{n}}\log\Bigg{(}\frac{d-b}{L_{n}}\Bigg{)},

where $L_{n}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{(b,d)\in\tilde{\xi}_{0,n}}d-b$ . We may represent $E_{n}$ as

\displaystyle-L_{n}^{-1}\sum_{(b,d)\in\tilde{\xi}_{0,n}}(d-b)\log(d-b)+\log L_{n}.

Another nontrivial statistic of interest is the ALPS statistic, defined in Thomas et al. (2023) and utilized in Thomas et al. (2023), Crozier et al. (2024), and Thomas et al. (2024). Its representation is

A(X_{1},\dots,X_{n})\equiv A_{n}\mathrel{\mathop{\mathchar 58\relax}}=\int_{0}^{\infty}\log\xi_{0,n}(\Delta_{\ell})\,\operatorname{d\!}{\ell},

and we define a truncation of the ALPS statistic as $A_{n}^{L}\mathrel{\mathop{\mathchar 58\relax}}=\int_{0}^{L}\log\xi_{0,n}(\Delta_{\ell})\,\operatorname{d\!}{\ell}.$ Before continuing, let us define $f_{e}(b,d)=(d-b)\log(d-b)$ and $f_{I}(b,d)=d-b$ . Both $f_{e}+1$ and $f_{I}$ are continuous, coercive, and largely nondecreasing in $d-b$ .

Corollary 3.10.

Assuming the conditions of Theorems 3.1 and 3.8, we have that

E_{n}-\log\tilde{\xi}_{0,n}(\Delta)\to\frac{\tilde{\xi}_{0}(f_{e})}{\tilde{\xi}_{0}(f_{I})}+\log\tilde{\xi}_{0}(f_{I}),\ \ \mathrm{a.s.},

and for any $L>0$ with $\xi_{0}(\Delta_{L})>0$ we have

L\log\xi_{0,n}(\Delta)-A_{n}^{L}\to-\int^{L}_{0}\log\xi_{0}(\Delta_{\ell})\operatorname{d\!}{\ell},\ \ \mathrm{a.s.},

as $n\to\infty$ . That is, the sublevel set persistent entropy and the ALPS statistic of a stationary and ergodic process converge almost surely.

Proof.

The proof follows fairly simply from Theorem 3.8. We know that

E_{n}=\frac{-\tilde{\xi}_{0,n}(f_{e}+1)+\tilde{\xi}_{0,n}(\Delta)}{\tilde{\xi}_{0,n}(f_{I})}+\log\tilde{\xi}_{0,n}(f_{I}).

Subtracting $\log\tilde{\xi}_{0,n}(\Delta)$ and applying Theorem 3.8 yields a limit of

\frac{-\tilde{\xi}_{0}(f_{e}+1)+1}{\tilde{\xi}_{0}(f_{I})}+\log\tilde{\xi}_{0}(f_{I}),

which finishes the proof, as $\tilde{\xi}_{0}$ a probability measure. For the ALPS statistic, we see that

L\log\xi_{0,n}(\Delta)-A_{n}^{L}=\int^{L}_{0}\log\bigg{(}\frac{\xi_{0,n}(\Delta)}{\xi_{0,n}(\Delta_{\ell})}\bigg{)}\operatorname{d\!}{\ell}.

If we fix a positive $\epsilon<\xi_{0}(\Delta_{L})$ , Corollary 3.7 implies that for $n\geq N(\omega)$ ( $N$ depending on the sample point $\omega\in\Omega$ ), we have

\log\bigg{(}\frac{\xi_{0,n}(\Delta)}{\xi_{0,n}(\Delta_{\ell})}\bigg{)}\leq-\log(\xi_{0}(\Delta_{\ell})-\epsilon)\leq-\log(\xi_{0}(\Delta_{L})-\epsilon),

for all $\ell\in[0,L]$ . Therefore, the bounded convergence assumption holds for all $\omega\in\Omega$ such that convergence holds. Hence, our result follows almost surely. ∎

Having demonstrated our strong law of large numbers for persistence diagrams, and its ramifications, we now turn our attention to the central limit theorem.

4. Central limit theorem

In this section, we prove a central limit theorem for the integral $\xi_{0,n}(f)$ , where $f$ is a step function. This follows from proving a CLT for linear combinations of persistent Betti numbers $\beta^{s,t}_{0,n}$ using the Lindeberg method for weakly dependent triangular arrays given in Neumann (2013). The desired result will follow as a consequence of demonstrating

n^{-1/2}\sum_{l=1}^{m}a_{l}\Big{(}\beta_{0,n}^{s_{l},t_{l}}-\mathbb{E}[\beta_{0,n}^{s_{l},t_{l}}]\Big{)}.

obeys a central limit theorem when $\mathcal{X}_{n}$ obeys weak dependence conditions (to be specified below) and $a_{1},\dots,a_{m}$ are arbitrary real numbers. The reason for this is that if $R_{l}=(s_{1},s_{2}]\times(t_{1},t_{2}]$ then

\mathbf{1}_{R_{l}}=\mathbf{1}_{(-\infty,s_{2}]\times(t_{1},\infty]}-\mathbf{1}_{(-\infty,s_{2}]\times(t_{2},\infty]}-\mathbf{1}_{(-\infty,s_{1}]\times(t_{1},\infty]}+\mathbf{1}_{(-\infty,s_{1}]\times(t_{2},\infty]}.

The Crámer-Wold device also provides us with finite-dimensional weak convergence as an added benefit.

As for the aforementioned notions of weak dependence, the one we employ is that of $\rho$ -mixing. To begin, note that for any two sub- $\sigma$ algebras $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ we define

\rho(\mathcal{A},\mathcal{B})\mathrel{\mathop{\mathchar 58\relax}}=\sup_{\begin{subarray}{c}X\in L^{2}(\mathcal{A})\,Y\in L^{2}(\mathcal{B})\end{subarray}}\big{|}\text{Corr}(X,Y)\big{|},

where $L^{2}(\mathcal{A})$ (resp. $L^{2}(\mathcal{B})$ ) is the space of square-integrable $\mathcal{A}$ -measurable (resp. $\mathcal{B}$ -measurable) random variables⁴⁴4For random variables $X,Y$ the value $\text{Corr}(X,Y)=\text{Cov}(X,Y)/\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}$ .. Furthermore, we define

\rho_{\mathcal{X}}(k)\mathrel{\mathop{\mathchar 58\relax}}=\sup_{m\in\mathbb{N}}\rho\big{(}\sigma(X_{1},\dots,X_{m}),\sigma(X_{m+k},X_{m+k+1},\dots)\big{)},

so that the stochastic process $\mathcal{X}=(X_{1},X_{2},\dots)$ is said to be $\rho$ -mixing if $\rho_{\mathcal{X}}(k)\to 0$ as $k\to\infty$ . For our limit theorems, we will require that $\sum_{k=1}^{\infty}\rho_{\mathcal{X}}(k)<\infty$ , which implies $\rho$ -mixing. More details on $\rho$ -mixing and other mixing conditions can be seen in Bradley (2005). Another particularly important condition for our proofs is that our stationary process obeys a certain condition on the probability distributions of the partial maxima decaying sufficiently quickly. This serves to limit any percolation-esque phenomena that would preclude a central limit theorem.

Definition 4.1.

A stationary stochastic process $\mathcal{X}=(X_{1},X_{2},\dots)$ with marginal distribution function $F$ is said to be max-root summable if for all $t$ with $F(t)<1$ we have

\sum_{i=1}^{\infty}i\sqrt{\mathbb{P}(X_{1}\leq t,\dots,X_{i}\leq t)}<\infty.

Before stating our main theorem, we will establish conditions on the stochastic process that guarantee max-root summability.

Proposition 4.2.

Suppose that $\mathcal{X}$ is a stationary stochastic process. If there is some $\epsilon>0$ s.t.

\mathbb{P}(X_{1}\leq t,\dots,X_{n}\leq t)=O(n^{-4-\epsilon}),

for all $t$ with $F(t)<1$ , then $\mathcal{X}$ is max-root summable.

Proof.

If the condition above holds there is some $C_{t}$ such that

n\sqrt{\mathbb{P}(X_{1}\leq t,\dots,X_{n}\leq t)}\leq\sqrt{C_{t}}n^{-1-\epsilon/2},

the right-hand side of which is clearly summable. ∎

Example 4.3.

Suppose that $\mathcal{X}$ is a (stationary) Markov chain with transition kernel $P$ such that for every $t$ with $F(t)<1$ there is some $\eta_{t}>0$ that satisifies

\sup_{x\leq t}P\big{(}x,(-\infty,t]\big{)}\leq 1-\eta_{t}.

By Theorem 3.4.1 in Meyn and Tweedie (2009), we have that

	$\displaystyle\mathbb{P}(X_{1}\leq t,\dots,X_{n}\leq t)$	$\displaystyle=\int_{x_{1}\leq t}\cdots\int_{x_{n-1}\leq t}F(\operatorname{d\!}x_{1})P(x_{1},\operatorname{d\!}x_{2})\cdots P(x_{n-2},\operatorname{d\!}x_{n-1})P\big{(}x_{n-1},(-\infty,t]\big{)}$
		$\displaystyle\leq\int_{x_{1}\leq t}\cdots\int_{x_{n-1}\leq t}F(\operatorname{d\!}x_{1})P(x_{1},\operatorname{d\!}x_{2})\cdots P(x_{n-2},\operatorname{d\!}x_{n-1})(1-\eta_{t}).$

Therefore, induction furnishes that

\mathbb{P}(X_{1}\leq t,\dots,X_{n}\leq t)\leq F(t)(1-\eta_{t})^{n-1},

and the condition in Proposition 4.2 can be simply established.

Example 4.4.

Suppose that $\mathcal{X}$ is stationary and $m$ -dependent, i.e. $\psi_{\mathcal{X}}(k)=0$ for all $k\geq m+1$ . Then we have

	$\displaystyle\mathbb{P}(X_{1}\leq t,\dots,X_{n}\leq t)$	$\displaystyle\leq\mathbb{P}(X_{1}\leq t,X_{m+2}\leq t,\dots,X_{\lfloor\frac{n-1}{m+1}\rfloor(m+1)+1}\leq t)$
		$\displaystyle=F(t)^{\lfloor\frac{n-1}{m+1}\rfloor+1}.$

Because $F(t)=0$ establishes max-root summability trivially, we take $0<F(t)<1$ . Then as $(\lfloor\frac{n-1}{m+1}\rfloor+1)\log\big{[}1/F(t)\big{]}\geq k\log n$ for any $k>0$ and $n$ large enough, then the condition in Proposition 4.2 is established.

To establish our CLT (Theorem 4.6 below), we first need to assess the limiting behavior of the covariance.

Proposition 4.5.

Let $\mathcal{X}$ be a stationary stochastic process that is max-root summable and satisfies $\sum_{k=1}^{\infty}\rho_{\mathcal{X}}(k)<\infty$ . Assume further that the marginal distribution of $X_{i}$ is continuous with distribution $F$ . Suppose that $-\infty<s_{i}\leq t_{i}\leq\infty$ for $i=1,2$ with $F(s_{1}\wedge s_{2})>0$ and $F(t_{1}\vee t_{2})<1$ .

	$\displaystyle\lim_{n\to\infty}n^{-1}\mathrm{Cov}\Big{(}\beta_{0,n}^{s_{1},t_{1}},\beta_{0,n}^{s_{2},t_{2}}\Big{)}$
	$\displaystyle\qquad\qquad=\mathrm{Cov}\big{(}Y_{2}^{\infty}(s_{1},t_{1}),Y_{2}^{\infty}(s_{2},t_{2})\big{)}$
	$\displaystyle\qquad\qquad+\sum_{k=1}^{\infty}\bigg{[}\mathrm{Cov}\big{(}Y_{2}^{\infty}(s_{1},t_{1}),Y_{2+k}^{\infty}(s_{2},t_{2})\big{)}+\mathrm{Cov}\big{(}Y_{2+k}^{\infty}(s_{1},t_{1}),Y_{2}^{\infty}(s_{2},t_{2})\big{)}\bigg{]}.$

where the terms $Y_{j}^{\infty}(s,t)$ are defined at (4) respectively.

With this all at hand, we may finally state the central limit theorem.

Theorem 4.6.

Let $\mathcal{X}$ be a stationary stochastic process that is max-root summable and satisfies $\sum_{k=1}^{\infty}\rho_{\mathcal{X}}(k)<\infty$ . Assume further that the marginal distribution of $X_{i}$ is continuous with distribution $F$ . Then for any function $f=\sum_{l=1}^{m}a_{l}\mathbf{1}_{R_{l}}$ with $a_{l}\in\mathbb{R}$ and $R_{l}\in\mathcal{R}$ , $l=1,\dots,m$ , if the corners $(s,t)$ of the rectangles satisfy $F(s)>0$ and $F(t)<1$ we have:

n^{-1/2}\big{(}\xi_{0,n}(f)-\mathbb{E}[\xi_{0,n}(f)]\big{)}\Rightarrow N(0,I_{f}),

and if each of the coordinates of $R_{l}$ lie in $\mathbb{R}$ for $l=1,\dots,m$ then

n^{-1/2}\big{(}\tilde{\xi}_{0,n}(f)-\mathbb{E}[\tilde{\xi}_{0,n}(f)]\big{)}\Rightarrow N(0,I_{f}),

as $n\to\infty$ , where $I_{f}$ is a nonnegative constant depending on $f$ .

We defer the proof to Section 6.

5. Discussion

In this paper, we have demonstrated a strong law of large numbers for a large class of integrals with the respect to the random measure induced by the $0^{th}$ sublevel set persistent homology of general stationary and ergodic processes. We also proved a central limit theorem for the same random measure for a large class of step functions. As the SLLNs—by consideration of the negated process $-X_{1},-X_{2},\dots$ —also pertain to superlevel sets, it would be interesting to consider the limiting behavior of the persistent homology of the extremes of a stationary stochastic process; the reason is due to the natural connection between the superlevel set value $\beta_{0,n}^{u_{n}(\tau),u_{n}(\tau)}$ (number of connected components above levels $u_{n}(\tau)$ , $\tau\geq 0$ ) and the clusters of exceedances seen in the extreme value theory literature (see chapter 6 of Kulik and Soulier, 2020).

Two potential improvements for this paper seem to lie in the weakening of conditions and the augmentation of the class of functions for which the central limit theorem holds (Theorem 4.6). There are likely only improvements to be made in the latter case, as the $\sum_{k=1}^{\infty}\rho_{\mathcal{X}}(k)<\infty$ condition is only slightly stronger than the slowest mixing rate of $\sum_{k=1}^{\infty}k^{-1}\rho_{\mathcal{X}}(k)<\infty$ for a conventional CLT to hold for a stationary sequence (Bradley, 1987). The improvement of the second objective seemingly depends on a more precise treatment of the covariance in Proposition 4.5, which is rather tedious as it stands. Nonetheless, such improvements would see utility as the class of functions of persistence diagrams used in practice are large, which is what motivated Section 3.1 (and this paper) to begin with. Expanding the CLT results to a functional CLT for the persistent Betti numbers (as in Krebs and Hirsch, 2022) may yield some progress towards this end, but we leave all the pursuits mentioned in these last two paragraphs for future work.

6. Central limit theorem proof

For the proof of our central limit theorem, we will employ Theorem 2.1 from Neumann (2013), which establishes a CLT for potentially nonstationary weakly dependent triangular arrays. As mentioned at the beginning of Section 4, it is sufficient to show that

(11)

n^{-1/2}\sum_{l=1}^{m}a_{l}\bigg{(}\beta_{0,n}^{s_{l},t_{l}}-\mathbb{E}[\beta_{0,n}^{s_{l},t_{l}}]\bigg{)},

converges to a Gaussian distribution for each $a_{1},\dots,a_{m}\in\mathbb{R}$ , to establish our desired convergence. Recall that at (4) we defined the indicator (Bernoulli) random variable $Y^{m}_{j,n}(s,t)$ and on the following line we noticed that

\beta_{0,n}^{s,t}=\sum_{j=1}^{n}Y^{n-j+1}_{j,n}(s,t),

so that (11) is equal to

n^{-1/2}\sum_{j=1}^{n}\sum_{l=1}^{m}a_{l}\bigg{(}Y_{j,n}^{n-j+1}(s_{l},t_{l})-\mathbb{E}[Y_{j,n}^{n-j+1}(s_{l},t_{l})]\bigg{)}.

For the proof the CLT it is convenient for us to establish first a CLT for a truncated version of the persistent Betti numbers—as was done in the proof of the Betti number CLT for the critical regime in the geometric setting, in Theorem 4.1 of Owada and Thomas (2020). Define first

\beta_{0,n,K}^{s,t}=\sum_{j=1}^{n}Y^{(n-j+1)\wedge K}_{j,n}(s,t)

Therefore, if we define

W_{j,n}\mathrel{\mathop{\mathchar 58\relax}}=n^{-1/2}\sum_{l=1}^{m}a_{l}\bigg{(}Y_{j,n}^{(n-j+1)\wedge K}(s_{l},t_{l})-\mathbb{E}[Y_{j,n}^{(n-j+1)\wedge K}(s_{l},t_{l})]\bigg{)},

establishing Theorem 4.6 amounts to establishing a CLT for $\sum_{j=1}^{n}W_{j,n}$ for each $K$ then showing that the difference between $\beta_{0,n}^{s,t}$ and $\beta_{0,n,K}^{s,t}$ disappears in probability. We will now quote the theorem which we will use to establish this.

Theorem 6.1 (Theorem 2.1 in Neumann, 2013).

Suppose that $(W_{j,n})_{j=1}^{n}$ with $n\in\mathbb{N}$ is a triangular array of random variables with $\mathbb{E}[W_{j,n}]=0$ for all $j,n$ and $\sup_{n}\sum_{j=1}^{n}\mathbb{E}[W_{j,n}^{2}]\leq M$ for some $M<\infty$ . Suppose further that

(12)

\mathrm{Var}\bigg{(}\sum_{j=1}^{n}W_{j,n}\bigg{)}\to\sigma^{2},\quad n\to\infty,

for some $\sigma^{2}\geq 0$ , and that for every $\epsilon>0$ we have

(13)

\sum_{k=1}^{n}\mathbb{E}[W_{j,n}^{2}\mathbf{1}\big{\{}|W_{j,n}|>\epsilon\big{\}}]\to 0,\quad n\to\infty.

Furthermore, assume that there exists a summable sequence of $\theta_{r}$ , $r\in\mathbb{N}$ , such that for all $q\in N$ and indices $1\leq u_{1}<u_{2}<\cdots<u_{q}+r=v_{1}\leq v_{2}\leq n$ , the following upper bounds for covariances hold true:

(14)

\Big{|}\mathrm{Cov}\big{(}g(W_{u_{1},n},\dots,W_{u_{q},n})W_{u_{q},n},W_{v_{1},n}\big{)}\Big{|}\leq\theta_{r}\big{(}\mathbb{E}[W_{u_{q},n}^{2}]+\mathbb{E}[W_{v_{1},n}^{2}]+n^{-1}\big{)}

and

(15)

\Big{|}\mathrm{Cov}\big{(}g(W_{u_{1},n},\dots,W_{u_{q},n}),W_{v_{1},n}W_{v_{2},n}\big{)}\Big{|}\leq\theta_{r}\big{(}\mathbb{E}[W_{v_{1},n}^{2}]+\mathbb{E}[W_{v_{2},n}^{2}]+n^{-1}\big{)}

for all measurable $g\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{q}\to\mathbb{R}$ with $\sup_{x\in\mathbb{R}^{q}}|g(x)|\leq 1$ . Then

\sum_{j=1}^{n}W_{j,n}\Rightarrow N(0,\sigma^{2}),\quad n\to\infty.

Proof of Theorem 4.6.

The finite-dimensional CLT proof for $\beta_{0,n,K}^{s,t}$ follows by checking that the conditions of Theorem 6.1 hold for our setup. First, we notice that

	$\displaystyle W_{j,n}^{2}=n^{-1}\sum_{l_{1}=1}^{m}\sum_{l_{2}=1}^{m}a_{l_{1}}a_{l_{2}}\bigg{(}Y_{j,n}^{(n-j+1)\wedge K}(s_{l_{1}},t_{l_{1}})-\mathbb{E}[Y_{j,n}^{(n-j+1)\wedge K}(s_{l_{1}},t_{l_{1}})]\bigg{)}$
	$\displaystyle\phantom{W_{j,n}^{2}=n^{-1}\sum_{l_{1}=1}^{m}\sum_{l_{2}=1}^{m}a_{l_{1}}a_{l_{2}}}\times\bigg{(}Y_{j,n}^{(n-j+1)\wedge K}(s_{l_{2}},t_{l_{2}})-\mathbb{E}[Y_{j,n}^{(n-j+1)\wedge K}(s_{l_{2}},t_{l_{2}})]\bigg{)},$

so that

\mathbb{E}[W_{j,n}^{2}]=n^{-1}\sum_{l_{1}=1}^{m}\sum_{l_{2}=1}^{m}a_{l_{1}}a_{l_{2}}\text{Cov}\Big{(}Y_{j,n}^{(n-j+1)\wedge K}(s_{l_{1}},t_{l_{1}}),Y_{j,n}^{(n-j+1)\wedge K}(s_{l_{2}},t_{l_{2}})\Big{)},

which is bounded above by

n^{-1}\bigg{(}\sum_{l=1}^{m}|a_{l}|\sqrt{\mathrm{Var}(Y_{j,n}^{(n-j+1)\wedge K}(s_{l},t_{l})})\bigg{)}^{2}\leq Mn^{-1},

for $M\mathrel{\mathop{\mathchar 58\relax}}=(\sum_{l}|a_{l}|)^{2}$ by the inequalities $|\text{Cov}(X,Y)|\leq\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}$ and $\mathrm{Var}(\mathbf{1}_{A})\leq\mathbb{P}(A)\leq 1$ . Thus, $\sup_{n}\sum_{j=1}^{n}\mathbb{E}[W_{j,n}^{2}]\leq M<\infty$ . If we note that

\mathrm{Var}\bigg{(}\sum_{j=1}^{n}W_{j,n}\bigg{)}=n^{-1}\sum_{l_{1}=1}^{m}\sum_{l_{2}=1}^{m}a_{l_{1}}a_{l_{2}}\text{Cov}\Big{(}\beta_{0,n,K}^{s_{l_{1}},t_{l_{1}}},\beta_{0,n,K}^{s_{l_{2}},t_{l_{2}}}\Big{)},

then $\mathrm{Var}\big{(}\sum_{j=1}^{n}W_{j,n}\big{)}$ converges to some limit $\sigma^{2}$ via arguments analogous to and much simpler than those of Proposition 4.5. Thus, (12) is satisfied. If we use the triangle inequality, we can see that for each $j$

|W_{j,n}|\leq 2n^{-1/2}\sum_{l=1}^{m}|a_{l}|=2(Mn)^{-1/2}

using the trivial indicator random variable bound $\mathbf{1}_{A}\leq 1$ . Therefore when $n\geq 4(M/\epsilon)^{2}$ we have that $\mathbf{1}\big{\{}|W_{j,n}|>\epsilon\big{\}}=0$ so that (13) holds as well. To finish the proof, we must show that (14) and (15) hold in Theorem 6.1 above. For both situations, we can ignore the case for $r\leq K+1$ , as we can set $\theta_{r}$ arbitrarily large in this case to get the required bounds in this case. Therefore, suppose that $r>K+1$ , so that $W_{u_{q},n}$ only depends on indices up to $u_{q}+K$ and $W_{v_{1},n}$ only depends on indices starting at $v_{1}-1=u_{q}+r-1>u_{q}+K$ .

We will only demonstrate (14), as (15) follows by a similar, simpler argument. For a fixed set of indices $u_{1},\dots,u_{q}$ and fixed $n$ let us denote $G\mathrel{\mathop{\mathchar 58\relax}}=g(W_{u_{1},n},\dots,W_{u_{q},n})$ . By the bilinearity of covariance, it will suffice to establish the required bounds in (14) for a single summand in

	$\displaystyle n^{-1}\sum_{l_{1}=1}^{m}\sum_{l_{2}=1}^{m}a_{l_{1}}a_{l_{2}}\text{Cov}\Big{(}G\big{\{}Y_{u_{q},n}^{(n-u_{q}+1)\wedge K}(s_{l_{1}},t_{l_{1}})-\mathbb{E}[Y_{u_{q},n}^{(n-u_{q}+1)\wedge K}(s_{l_{1}},t_{l_{1}})]\big{\}},$
(16)		$\displaystyle\phantom{n^{-1}\sum_{l_{1}=1}^{m}\sum_{l_{2}=1}^{m}a_{l_{1}}a_{l_{2}}\text{Cov}\Big{(}}Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l_{2}},t_{l_{2}})-\mathbb{E}[Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l_{2}},t_{l_{2}})]\Big{)}$	.

provided that such a bound is uniform in $l_{1},l_{2}$ . It can be shown that the covariance term in (16) is equal to

	$\displaystyle\text{Cov}\Big{(}GY_{u_{q},n}^{(n-u_{q}+1)\wedge K}(s_{l_{1}},t_{l_{1}}),Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l_{2}},t_{l_{2}})\Big{)}$
(17)		$\displaystyle\qquad\qquad\qquad-\mathbb{E}[Y_{u_{q},n}^{(n-u_{q}+1)\wedge K}(s_{l_{1}},t_{l_{1}})]\text{Cov}(G,Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l_{2}},t_{l_{2}})),$

and the absolute value of (17) can be bounded above by

|\text{Cov}\Big{(}GY_{u_{q},n}^{(n-u_{q}+1)\wedge K}(s_{l_{1}},t_{l_{1}}),Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l_{2}},t_{l_{2}})\Big{)}|+|\text{Cov}(G,Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l_{2}},t_{l_{2}}))|.

Because $GY_{u_{q},n}^{(n-u_{q}+1)\wedge K}(s_{l_{1}},t_{l_{1}})=g^{*}(W_{u_{1},n},\dots,W_{u_{q},n})$ , for $g^{*}$ measurable and $\sup_{x}|g^{*}(x)|\leq 1$ , the required bound will follow provided we find a suitable bound for the quantity $|\text{Cov}(G,Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l},t_{l}))|$ . By definition of $\rho$ -mixing and the trivial bound $\mathrm{Var}(X)\leq E[X^{2}]$ we have

|\text{Cov}(G,Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l},t_{l}))|\leq\rho_{\mathcal{X}}(r-1-K)\sqrt{\mathrm{Var}(G)}\sqrt{\mathrm{Var}(Y_{v_{1},n}^{(n-v_{1}+1)\wedge K}(s_{l},t_{l}))}\leq\rho_{\mathcal{X}}(r-1-K).

By assumption, $\sum_{r>K+1}\rho_{\mathcal{X}}(r-1-K)<\infty$ so that (14) is established. As alluded to earlier, the proof for (15) follows in exactly the same way, hence

(18)

n^{-1/2}\sum_{l=1}^{m}a_{l}\bigg{(}\beta_{0,n,K}^{s_{l},t_{l}}-\mathbb{E}[\beta_{0,n,K}^{s_{l},t_{l}}]\bigg{)}\Rightarrow N(0,\sigma_{K}^{2}),\quad n\to\infty

for all $K\in\mathbb{N}$ . As the dominated convergence assumption holds true in Proposition 4.5, it is straightforward to see that $\sigma_{K}^{2}\to\sigma^{2}$ as $K\to\infty$ , where $\sigma^{2}$ is the limiting variance of (11). Hence $N(0,\sigma_{K}^{2})\Rightarrow N(0,\sigma^{2})$ as $K\to\infty$ as well (using Lévy’s continuity theorem, for example). Theorem 3.2 in Billingsley (1999) will yield the rest if we can show that

\lim_{K\to\infty}\limsup_{n\to\infty}\mathbb{P}(|Z_{n,K}-Z_{n}|\geq\epsilon)=0,

where $Z_{n}$ is the sum of persistent Betti numbers in (11) and $Z_{n,K}$ is the $K$ -truncated version on the left-hand side of (18). An application of Chebyshev’s inequality and the covariance inequality yields

	$\displaystyle\mathbb{P}(\|Z_{n}-Z_{n,K}\|\geq\epsilon)$	$\displaystyle\leq\frac{\mathbb{E}\big{\|}Z_{n}-Z_{n,K}\big{\|}^{2}}{\epsilon^{2}}$
		$\displaystyle=\frac{1}{\epsilon^{2}n}\mathrm{Var}\bigg{(}\sum_{l=1}^{m}a_{l}\Big{[}\beta_{0,n}^{s_{l},t_{l}}-\beta_{0,n,K}^{s_{l},t_{l}}\Big{]}\bigg{)}$
(19)			$\displaystyle\leq\frac{1}{\epsilon^{2}}\bigg{(}\sum_{l=1}^{m}a_{l}\sqrt{n^{-1}\mathrm{Var}\big{(}\beta_{0,n}^{s_{l},t_{l}}-\beta_{0,n,K}^{s_{l},t_{l}}\big{)}}\Bigg{)}^{2}.$

The quantity $n^{-1}\mathrm{Var}\big{(}\beta_{0,n}^{s_{l},t_{l}}-\beta_{0,n,K}^{s_{l},t_{l}}\big{)}$ converges to a limit defined by the terms (24), (25), and (26) below with the restriction that $i_{1},i_{2}>K$ . As each of the sums in (24), (25), and (26) are absolutely convergent, their restriction with $i_{1},i_{2}>K$ tends to 0 as $K\to\infty$ , and the CLT follows.

Finally, for any $A\in\mathcal{B}(\tilde{\Delta})$

\xi_{0,n}(A)=\tilde{\xi}_{0,n}(A)

so that if each coordinate of $R_{l}$ is in $\mathbb{R}$ , then $R_{l}\in\mathcal{B}(\tilde{\Delta})$ and the result is proved for the restricted persistence diagram as well. ∎

We finish this section with a proof of the limiting covariance seen in Proposition 4.5, which we will break into a few lemmas.

Proof of Proposition 4.5. Let us define

C^{n}_{i,j}(s,t)\mathrel{\mathop{\mathchar 58\relax}}=\mathbf{1}\bigg{\{}\bigvee_{k=j}^{j+i-1}X_{k,n}\leq t,\bigwedge_{k=j}^{j+i-1}X_{k,n}\leq s\bigg{\}}\mathbf{1}\big{\{}X_{j-1,n}\wedge X_{j+i,n}>t\big{\}},

where $C_{i,j}(s,t)\equiv C^{\infty}_{i,j}(s,t)$ is analogously defined for the entire sequence $\mathcal{X}$ . Therefore we have

\beta_{0,n}^{s,t}=\sum_{j=1}^{n}\sum_{i=1}^{n-j+1}C^{n}_{i,j}(s,t)=\sum_{i=1}^{n}\sum_{j=1}^{n-i+1}C^{n}_{i,j}(s,t).\\

Thus, it follows that

	$\displaystyle\mathrm{Cov}\Big{(}\beta_{0,n}^{s_{1},t_{1}},\beta_{0,n}^{s_{2},t_{2}}\Big{)}$
	$\displaystyle\qquad=\mathbb{E}\big{[}\beta_{0,n}^{s_{1},t_{1}}\beta_{0,n}^{s_{2},t_{2}}\big{]}-\mathbb{E}[\beta_{0,n}^{s_{1},t_{1}}]\mathbb{E}[\beta_{0,n}^{s_{2},t_{2}}]$
(20)		$\displaystyle\qquad=\sum_{i_{1},i_{2}}\sum_{j_{1},j_{2}}\mathbb{E}[C^{n}_{i_{1},j_{1}}(s_{1},t_{1})C^{n}_{i_{2},j_{2}}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},j_{1}}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},j_{2}}(s_{2},t_{2})],$

where $i_{1},i_{2}=1,\dots,n$ with $j_{1}=1,\dots,n-i_{1}+1$ , and $j_{2}=1,\dots,n-i_{2}+1$ . We may then break (20) into

	$\displaystyle\sum_{i_{1},i_{2}}\sum_{j=1}^{n-i_{1}\vee i_{2}+1}\mathbb{E}[C^{n}_{i_{1},j}(s_{1},t_{1})C^{n}_{i_{2},j}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},j}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},j}(s_{2},t_{2})]$
	$\displaystyle\qquad+\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{2}}\sum_{j=1}^{n-i_{1}\vee(i_{2}+k)+1}\mathbb{E}[C^{n}_{i_{1},j}(s_{1},t_{1})C^{n}_{i_{2},j+k}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},j}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},j+k}(s_{2},t_{2})]$
(21)		$\displaystyle\qquad+\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{1}}\sum_{j=1}^{n-i_{2}\vee(i_{1}+k)+1}\mathbb{E}[C^{n}_{i_{1},j+k}(s_{1},t_{1})C^{n}_{i_{2},j}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},j+k}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},j}(s_{2},t_{2})].$

For now, we will exclude the boundary terms from each sum—which use $X_{0,n}$ and $X_{n+1,n}$ . We will treat the boundary terms later. The nonboundary terms of the expression (21) can thus be simplified based on the assumed stationarity of $\mathcal{X}_{n}$ to be

	$\displaystyle\sum_{i_{1},i_{2}}(n-i_{1}\vee i_{2}-1)\Big{(}\mathbb{E}[C^{n}_{i_{1},2}(s_{1},t_{1})C^{n}_{i_{2},2}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},2}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},2}(s_{2},t_{2})]\Big{)}$
	$\displaystyle\ +\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{2}}(n-i_{1}\vee(i_{2}+k)-1)\Big{(}\mathbb{E}[C^{n}_{i_{1},2}(s_{1},t_{1})C^{n}_{i_{2},2+k}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},2}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},2+k}(s_{2},t_{2})]\Big{)}$
(22)		$\displaystyle\ +\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{1}}(n-i_{2}\vee(i_{1}+k)-1)\Big{(}\mathbb{E}[C^{n}_{i_{1},2+k}(s_{1},t_{1})C^{n}_{i_{2},2}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},2+k}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},2}(s_{2},t_{2})]\Big{)}.$

Dividing by $n$ , we may express the first term in (22) as

(23)

\sum_{i_{1}=1}^{\infty}\sum_{i_{2}=1}^{\infty}(1-(i_{1}\vee i_{2}-1)/n)_{+}\Big{(}\mathbb{E}[C^{n}_{i_{1},2}(s_{1},t_{1})C^{n}_{i_{2},2}(s_{2},t_{2})]-\mathbb{E}[C^{n}_{i_{1},2}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},2}(s_{2},t_{2})]\Big{)}.

Assuming we can show that

\sum_{i_{1}=1}^{\infty}\sum_{i_{2}=1}^{\infty}\Big{|}\mathbb{E}[C_{i_{1},2}(s_{1},t_{1})C_{i_{2},2}(s_{2},t_{2})]-\mathbb{E}[C_{i_{1},2}(s_{1},t_{1})]\mathbb{E}[C_{i_{2},2}(s_{2},t_{2})]\Big{|}<\infty,

where we drop the superscript $n$ as mentioned at the start of Section 4, then (23) will converge to

(24)

\sum_{i_{1}=1}^{\infty}\sum_{i_{2}=1}^{\infty}\mathbb{E}[C_{i_{1},2}(s_{1},t_{1})C_{i_{2},2}(s_{2},t_{2})]-\mathbb{E}[C_{i_{1},2}(s_{1},t_{1})]\mathbb{E}[C_{i_{2},2}(s_{2},t_{2})].

Similarly, we will get limits of

(25)

\sum_{i_{1}=1}^{\infty}\sum_{i_{2}=1}^{\infty}\sum_{k=1}^{\infty}\mathbb{E}[C_{i_{1},2}(s_{1},t_{1})C_{i_{2},2+k}(s_{2},t_{2})]-\mathbb{E}[C_{i_{1},2}(s_{1},t_{1})]\mathbb{E}[C_{i_{2},2+k}(s_{2},t_{2})],

and

(26)

\sum_{i_{1}=1}^{\infty}\sum_{i_{2}=1}^{\infty}\sum_{k=1}^{\infty}\mathbb{E}[C_{i_{1},2+k}(s_{1},t_{1})C_{i_{2},2}(s_{2},t_{2})]-\mathbb{E}[C_{i_{1},2+k}(s_{1},t_{1})]\mathbb{E}[C_{i_{2},2}(s_{2},t_{2})],

for the second and third terms in (22), provided the dominated convergence assumption holds for each of these cases. In fact, these three sums comprise the limit of the covariance. However, to establish that, we must ensure that the “boundary terms” vanish, which we do in Lemma 6.3. A useful fact will aid in the proof of the covariance limit above and the lemma below.

Lemma 6.2.

Fix $k\geq 0$ . Suppose that $i_{2}+k>i_{1}$ and $k\leq i_{1}$ , then for any values of $t_{1},t_{2}$ we have

C^{n}_{i_{1},j}(s_{1},t_{1})C^{n}_{i_{2},j+k}(s_{2},t_{2})=0.

Analogously, if $i_{1}+k>i_{2}$ and $k\leq i_{2}$ , then for any values of $t_{1},t_{2}$ we have

C^{n}_{i_{1},j+k}(s_{1},t_{1})C^{n}_{i_{2},j}(s_{2},t_{2})=0.

Proof.

Note that if $i_{2}+k>i_{1}$ and $k\leq i_{1}$ , then it must be the case that there exists indices $l,l^{\prime}$ such that if $C_{i_{1},j}(s_{1},t_{1})C_{i_{2},j+k}(s_{2},t_{2})=1$ then

X_{l}\leq t_{1},X_{l}>t_{2}\text{ and }X_{l}^{\prime}>t_{1},X_{l}^{\prime}\leq t_{2},

a contradiction because $t_{1}>t_{2}$ and $t_{2}>t_{1}$ cannot simultaneously hold—even if $t_{1}=t_{2}$ . The proof for the second case follows by the same argument. ∎

Lemma 6.3.

If $\mathcal{X}$ is a $\rho$ -mixing stationary stochastic process that is max-root summable then the boundary terms in (21) are $o(n)$ as $n\to\infty$ .

Proof.

The boundary terms (21) comprise those terms in the first sum that satisfy $j=1$ or $j+(i_{1}\vee i_{2})=n+1$ , the terms in the second sum satisfying $j=1$ or $j+i_{1}\vee(i_{2}+k)=n+1$ , and the terms in the third sum satisfying $j=1$ , or $j+(i_{1}+k)\vee i_{2}=n+1$ . Thus, the boundary terms can be represented as

	$\displaystyle\sum_{i_{1},i_{2}}\text{Cov}\big{(}C^{n}_{i_{1},1}(s_{1},t_{1}),C^{n}_{i_{2},1}(s_{2},t_{2})\big{)}+\text{Cov}\big{(}C^{n}_{i_{1},n-i_{1}\vee i_{2}+1}(s_{1},t_{1}),C^{n}_{i_{2},n-i_{1}\vee i_{2}+1}(s_{2},t_{2})\big{)}$
	$\displaystyle+\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{2}}\text{Cov}\big{(}C^{n}_{i_{1},1}(s_{1},t_{1}),C^{n}_{i_{2},1+k}(s_{2},t_{2})\big{)}$
	$\displaystyle\qquad\qquad\qquad+\text{Cov}\big{(}C^{n}_{i_{1},n-i_{1}\vee(i_{2}+k)+1}(s_{1},t_{1}),C^{n}_{i_{2},n-(i_{1}-k)\vee i_{2}+1}(s_{2},t_{2})\big{)}$
	$\displaystyle+\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{1}}\text{Cov}\big{(}C^{n}_{i_{1},1+k}(s_{1},t_{1}),C^{n}_{i_{2},1}(s_{2},t_{2})\big{)}$
(27)		$\displaystyle\qquad\qquad\qquad+\text{Cov}\big{(}C^{n}_{i_{1},n-i_{1}\vee(i_{2}-k)+1}(s_{1},t_{1}),C^{n}_{i_{2},n-(i_{1}+k)\vee i_{2}+1}(s_{2},t_{2})\big{)}.$

We may bound the absolute value of the first sum in (27) by

	$\displaystyle\sum_{i_{1},i_{2}}\sqrt{\mathrm{Var}(C^{n}_{i_{1},1}(s_{1},t_{1}))}\sqrt{\mathrm{Var}(C^{n}_{i_{2},1}(s_{2},t_{2}))}$
	$\displaystyle\phantom{\sum_{i_{1},i_{2}}}\qquad\qquad+\sqrt{\mathrm{Var}(C^{n}_{i_{1},n-(i_{1}\vee i_{2})+1}(s_{1},t_{1}))}\sqrt{\mathrm{Var}(C^{n}_{i_{2},n-i_{1}\vee i_{2}+1}(s_{2},t_{2}))}$
	$\displaystyle\leq 2\sum_{i_{1},i_{2}}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}<\infty,$

and thus $o(n)$ —where we use the inequalities $|\text{Cov}(X,Y)|\leq\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}$ , $\mathrm{Var}(\mathbf{1}_{A})\leq\mathbb{P}(A)$ , and the fact that $\mathcal{X}$ is max-root summable. We now will finish the proof by showing that the second sum in (27) is $o(n)$ as well. That the third sum in (27) is $o(n)$ follows by an essentially symmetric proof. We may bound the absolute value of the second sum in (27) by

	$\displaystyle 2$	$\displaystyle\sum_{i_{1},i_{2}}\sum_{k=i_{1}+2}^{n-i_{2}}\rho_{\mathcal{X}}(k-i_{1}-1)\sqrt{\mathrm{Var}(C^{n}_{i_{1},1}(s_{1},t_{1}))}\sqrt{\mathrm{Var}(C^{n}_{i_{2},1}(s_{2},t_{2}))}$
	$\displaystyle+$	$\displaystyle\sum_{i_{1},i_{2}}\sum_{k=1}^{i_{1}+1}\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},1}(s_{1},t_{1}),C^{n}_{i_{2},1+k}(s_{2},t_{2})\big{)}\Big{\|}$
(28)			$\displaystyle\qquad\qquad\qquad+\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},n-i_{1}\vee(i_{2}+k)+1}(s_{1},t_{1}),C^{n}_{i_{2},n-(i_{1}-k)\vee i_{2}+1}(s_{2},t_{2})\big{)}\Big{\|}.$

The first sum in (28) follows from the definition of $\rho$ -mixing and the fact that $F(s_{i})>0$ and $F(t_{i})<1$ . Dividing the aforementioned first sum by $n$ we see that

	$\displaystyle 2n^{-1}\sum_{i_{1},i_{2}}\sum_{k=i_{1}+2}^{n-i_{2}}\rho_{\mathcal{X}}(k-i_{1}-1)\sqrt{\mathrm{Var}(C^{n}_{i_{1},1}(s_{1},t_{1}))}\sqrt{\mathrm{Var}(C^{n}_{i_{2},1}(s_{2},t_{2}))}$
	$\displaystyle\qquad\leq 2n^{-1}\sum_{k=1}^{n}\rho_{\mathcal{X}}(k)\sum_{i_{1},i_{2}}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}$

which tends to $0$ as $n\to\infty$ by max-root summability and the fact that $\rho_{\mathcal{X}}(k)\to 0$ . The second sum in (28) is a little more delicate. Before continuing, note that $\big{|}k\in\mathbb{N}\mathrel{\mathop{\mathchar 58\relax}}k>i_{1}-i_{2},\,k\leq i_{1}\big{|}=i_{1}\wedge i_{2}\leq i_{1}i_{2}$ when both terms are at least 1. Hence, Lemma 6.2 implies that the second sum in (28) equals

	$\displaystyle\sum_{i_{2}=1}^{n}\sum_{i_{1}=1}^{n}\sum_{k=1+(i_{1}-i_{2})_{+}}^{i_{1}}\mathbb{E}[C^{n}_{i_{1},1}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},1+k}(s_{2},t_{2})]$
	$\displaystyle\phantom{\quad+\sum_{i_{2}=1}^{n}\sum_{i_{1}=i_{2}+1}^{n}\sum_{k=1}^{i_{1}-i_{2}}}\qquad+\mathbb{E}[C^{n}_{i_{1},n-i_{1}\vee(i_{2}+k)+1}(s_{1},t_{1})]\mathbb{E}[C^{n}_{i_{2},n-(i_{1}-k)\vee i_{2}+1}(s_{2},t_{2})]$
	$\displaystyle\quad+\sum_{i_{2}=1}^{n}\sum_{i_{1}=1}^{n-i_{2}-1}\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},1}(s_{1},t_{1}),C^{n}_{i_{2},i_{1}+2}(s_{2},t_{2})\big{)}\Big{\|}+\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},n-(i_{1}+i_{2})}(s_{1},t_{1}),C^{n}_{i_{2},n-i_{2}+1}(s_{2},t_{2})\big{)}\Big{\|}$
(29)		$\displaystyle\quad+\sum_{i_{2}=1}^{n}\sum_{i_{1}=i_{2}+1}^{n}\sum_{k=1}^{i_{1}-i_{2}}\bigg{(}\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},1}(s_{1},t_{1}),C^{n}_{i_{2},1+k}(s_{2},t_{2})\big{)}\Big{\|}.$
	$\displaystyle\phantom{\quad+\sum_{i_{2}=1}^{n}\sum_{i_{1}=i_{2}+1}^{n}\sum_{k=1}^{i_{1}-i_{2}}}\qquad+\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},n-i_{1}+1}(s_{1},t_{1}),C^{n}_{i_{2},n-i_{1}+k+1}(s_{2},t_{2})\big{)}\Big{\|}\bigg{)}.$

We may bound the first term in (29)

	$\displaystyle 2\sum_{i_{2}=1}^{n}\sum_{i_{1}=1}^{n}i_{1}i_{2}\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})$
	$\displaystyle\qquad=2\sum_{i_{1}=1}^{n}i_{1}\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})\sum_{i_{2}=1}^{n}i_{2}\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})$
	$\displaystyle\qquad=o(n)$

by the max-root summability condition. Furthermore, we can bound the second sum in (29) by

2\sum_{i_{1}=1}^{n}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sum_{i_{2}=1}^{n}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}=o(n),

using the covariance inequality $\text{Cov}(X,Y)\leq\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}$ , and again using the max-root summability condition. Finally, we bound the third sum in (29) by

	$\displaystyle 2\sum_{i_{2}=1}^{n}\sum_{i_{1}=i_{2}+1}^{n}(i_{1}-i_{2})\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}$
	$\displaystyle\leq 2\sum_{i_{1}=1}^{n}i_{1}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sum_{i_{2}=1}^{n}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}$
	$\displaystyle=o(n),$

by a final application of the max-root summability condition. ∎

Having shown that the boundary terms vanish under our conditions, it will suffice to show the dominated convergence condition for the terms in (22) divided by $n$ , which will then tend to the sums of (24), (25), and (26) respectively. First, we divide each term by $n$ and see that the first covariance term with absolute summands is bounded above (using again the usual covariance inequalities) by

	$\displaystyle\sum_{i_{1},i_{2}}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}$
	$\displaystyle\qquad=\sum_{i_{1}}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sum_{i_{2}}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}<\infty,$

by applying max-root summability for each sum. We now prove the dominated convergence assumption for the second sum (divided by $n$ ) in (22), as the third sum follows an analogous proof. This procedure yields an upper bound of

	$\displaystyle\sum_{i_{1},i_{2}}\sum_{k=1}^{n-i_{2}}\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},2}(s_{1},t_{1})C^{n}_{i_{2},2+k}(s_{2},t_{2})\big{)}\Big{\|}$
	$\displaystyle\qquad\leq\sum_{i_{1},i_{2}}\sum_{k=1}^{i_{1}+1}\Big{\|}\text{Cov}\big{(}C^{n}_{i_{1},2}(s_{1},t_{1})C^{n}_{i_{2},2+k}(s_{2},t_{2})\big{)}\Big{\|}$
(30)		$\displaystyle\qquad+\sum_{i_{1},i_{2}}\sum_{k=i_{1}+2}^{n-i_{2}}\rho_{\mathcal{X}}(k-i_{1}-1)\sqrt{\mathrm{Var}(C^{n}_{i_{1},2}(s_{1},t_{1}))}\sqrt{\mathrm{Var}(C^{n}_{i_{2},2}(s_{2},t_{2}))}.$

The first sum in (30) we may bound by

\sum_{i_{1},i_{2}}(i_{1}+1)\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}<\infty,

by max-root summability of $\mathcal{X}$ . The second sum in (30) is bounded above by

		$\displaystyle\sum_{k=1}^{n}\rho_{\mathcal{X}}(k)\sum_{i_{1},i_{2}}\sqrt{\mathrm{Var}(C^{n}_{i_{1},1}(s_{2},t_{1}))}\sqrt{\mathrm{Var}(C^{n}_{i_{2},2}(s_{2},t_{2}))}$
	$\displaystyle\leq$	$\displaystyle\sum_{k=1}^{\infty}\rho_{\mathcal{X}}(k)\sum_{i_{1},i_{2}}\sqrt{\mathbb{P}(X_{1}\leq t_{1},\dots,X_{i_{1}}\leq t_{1})}\sqrt{\mathbb{P}(X_{1}\leq t_{2},\dots,X_{i_{2}}\leq t_{2})}<\infty.$

by assumption.

As for the representation of $\lim_{n\to\infty}n^{-1}\mathrm{Cov}\Big{(}\beta_{0,n}^{s_{1},t_{1}},\beta_{0,n}^{s_{2},t_{2}}\Big{)}$ , we note that the sums (24), (25), and (26) are all absolutely convergent, hence we may split the sums and apply the monotone convergence theorem to each, and recombine to get the stated representation. $\square$

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Convergence of persistence diagrams for discrete time stationary processes

Abstract.

1. Introduction

2. Background

2.1. Homology

2.2. Persistent homology and representations

2.3. Probability and persistence

Proposition 2.1.

Proof.

Lemma 2.2.

Proof.

Proposition 2.3.

Proof.

3. Strong law of large numbers

Theorem 3.1.

Proof.

Remark 3.2.

Proposition 3.3.

Proof.

Example 3.4.

Corollary 3.5.

Example 3.6.

Corollary 3.7.

3.1. SLLN for unbounded functions

Theorem 3.8.

Proof.

Corollary 3.9.

3.2. Strong law of large numbers: two examples

Corollary 3.10.

Proof.

4. Central limit theorem

Definition 4.1.

Proposition 4.2.

Proof.

Example 4.3.

Example 4.4.

Proposition 4.5.

Theorem 4.6.

5. Discussion

6. Central limit theorem proof

Theorem 6.1 (Theorem 2.1 in Neumann, 2013).

Proof of Theorem 4.6.

Lemma 6.2.

Proof.

Lemma 6.3.

Proof.

References

Convergence of persistence diagrams
for discrete time stationary processes