A large sample property in approximating the superposition of i.i.d. finite point processes

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Abstract

One of the main differences between the central limit theorem and the Poisson law of small numbers is that the former possesses the large sample property (LSP), i.e., the error of normal approximation to the sum of n independent identically distributed (i.i.d.) random variables converges to 0 as n. Since 1980s, considerable effort has been devoted to recovering the LSP for the law of small numbers in discrete random variable approximation. In this paper, we aim to establish the LSP for the superposition of i.i.d. finite point processes.

Introduction

The central limit theorem states that the distribution of the sum S:=i=1nXi of independent copies of a random variable X with finite second moment, after being normalized, converges weakly to the standard normal distribution. The Berry–Esseen bound ensures that, if X has the finite third moment, the error of the normal approximation, measured in the Kolmogorov metric, is not worse than cn, where c is a constant determined by the distribution of X. In other words, the central limit theorem has the large sample property (LSP), i.e., the quality of the approximation improves as the sample size becomes large. The LSP can also be established for the functional central limit theorem measured in the Lévy–Prokhorov distance [13], [25], [28], [33], [40]. Moreover, Stein’s method can be used to estimate the errors of diffusion approximation [2].

The Poisson law of small numbers, on the other hand, does not possess the LSP. More precisely, if Xi’s are independent indicator random variables with P(Xi=1)=1P(Xi=0)=pi for each i, then the total variation distance between the distribution of W=i=1nIi and the Poisson distribution with mean λi=1npi is of the order Ωλ1i=1npi2 [6]. In particular, if pi=p for all i, one can see that the quality of approximation does not improve when n becomes large. This is due to the fact that a Poisson distribution has only one parameter while a normal distribution has two parameters. To recover the LSP, one has to introduce more parameters into the approximating distributions, e.g., signed compound Poisson measures, translated Poisson, compound Poisson, negative binomial and polynomial birth–death distributions [4], [5], [12], [16], [18], [32], [35], [37], [41].

If we consider point processes rather than nonnegative integer-valued random variables, the counterpart is the superposition Ξ1++Ξn of point processes {Ξi:1in}. The pioneering work of Grigelionis [27] demonstrates that the distribution of the superposition of independent sparse point processes on the carrier space R+ converges weakly to a Poisson process distribution. The same phenomenon can be established for the superposition of dependent sparse point processes on a general carrier space [14], [26], [29], [30]. The accuracy of Poisson point process approximation has been of considerable interest since 1970s [14], [39]. Stein’s method for Poisson process approximation was subsequently established by [1], [3] for estimating the approximation errors and the method was further refined by [17], [19], [42]. In the context of the aforementioned superposition of i.i.d. point processes, no error estimates were studied until the last decade [20], [38] and these studies show that the Poisson point process approximation to the superposition of i.i.d. point processes does not possess the LSP either. The aim of this note is to show that, by introducing more parameters into the approximating point process distribution, it is possible to recover the LSP in approximating the superposition of i.i.d. finite point processes. It is worthwhile to mention that there exists a well-understood theory for limit theorems of superpositions of point processes on Euclidean spaces of any dimension [31, Chapter 4].

Given that a Poisson point process on a compact metric space can be viewed as a Poisson number Z of i.i.d. points in the space, a natural step of introducing more parameters into the approximating point process is to replace the Poisson number Z by a random variable N whose distribution is controlled by two or more parameters, such as the translated Poisson [5], [37], negative binomial [16] and polynomial birth–death distributions [18]. The family of approximating distributions we will consider in this note is the polynomial birth–death process distributions introduced in [44]. To quantify the difference between two point processes, as in [20], [38], we use the Wasserstein distance d2 initiated in [3]. The formal statement of the main result is given in Theorem 2.4. Several applications are provided in Section 3 to illustrate the order of convergence in the LSP. Section 4 is devoted to the proof of the main result.

Section snippets

Preliminaries and the main result

1. Point processes. For the reader’s convenience, in this part, we collect some basic concepts and facts, and introduce a partitional total variation distance for comparing point processes under a partition of the carrier space. The basic concepts needed for this note are point process, reduced palm process [30, Chapter 10], the Wasserstein distance d2 [3] and partition [45].

Let Γ be a compact metric space with metric d0 bounded by 1. Let (Γ) be the Borel σ-algebra induced by d0. A

Examples

In this section, we demonstrate the use of Theorem 2.4 in five applications: Bernoulli process, Bernoulli process with shifts, compound Poisson process, renewal process and entrances and exits of Markov process. For simplicity, except in Section 3.3, we only consider point processes on the carrier space Γ=[0,1] with d0(x,y)=|xy|. Extension to any compact carrier space is a straightforward exercise.

The proof of Theorem 2.4

For the ease of reading, we recap the main points of [45]. The advantage of using πa,b;β;μ as approximating distribution is that it can be considered as the unique stationary distribution of an -valued positive recurrent Markov point process with the generator Ah(ξ)(a+b|ξ|)Γ(h(ξ+δx)h(ξ))μ(dx)+(1+β(|ξ|1))Γ(h(ξδx)h(ξ))ξ(dx). We use Zξ() to stand for a birth–death point process with generator A and initial configuration ξ. For any bounded measurable function f on (,()), it can be shown

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We thank the anonymous referees for their suggestions, comments and questions that lead to the improved presentation.

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  • Cited by (1)

    1

    Work supported by a Research Training Program Scholarship, Australia and a Xing Lei Cross-Disciplinary Ph.D. Scholarship in Mathematics and Statistics at the University of Melbourne, Australia.

    2

    Work supported in part by the Belz fund, Australian Research Council Grants Nos. DP150101459 and DP190100613.

    3

    Work supported in part by NSF of China 11371040.

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