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arxiv: 1906.10008 · v1 · pith:KSGKZWUEnew · submitted 2019-06-24 · 🧮 math.PR

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

Tianshu Cong , Aihua Xia , Fuxi Zhang This is my paper

Pith reviewed 2026-05-25 16:56 UTC · model grok-4.3

classification 🧮 math.PR MSC 60G5560F05
keywords large sample propertysuperposition of point processesPoisson approximationi.i.d. processestotal variation distancelimit theorems for point processes
0
0 comments X

The pith

The approximation error for the superposition of n i.i.d. point processes to a Poisson process decreases with n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish the large sample property for the superposition of n independent identically distributed point processes, meaning the error in approximating their sum by a Poisson process shrinks as n grows. This mirrors the central limit theorem's improvement with sample size, in contrast to the usual Poisson law of small numbers where the error does not necessarily decrease. Proving the property would extend Poisson approximation results from the 1980s onward to large collections of processes. It focuses on recovering this decreasing-error behavior under regularity conditions on the processes.

Core claim

The paper establishes the large sample property for the superposition of n i.i.d. point processes: under regularity conditions, the error of the Poisson approximation decreases as a function of n.

What carries the argument

The large sample property (LSP) for superposition of i.i.d. point processes, which requires the approximation error to be a decreasing function of the number of summands n.

If this is right

  • The distance between the law of the superposition and the Poisson limit improves quantitatively as n grows.
  • Error bounds obtained via Stein's method or coupling become tighter for larger collections of processes.
  • The result applies directly to modeling the aggregate of many identical rare-event sources.
  • Convergence rates can be stated explicitly in terms of n for the total variation or other metrics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The property may support refined bounds when superpositions model high-volume systems such as queues or sensor networks.
  • Analogous large-sample improvements could be checked for compound Poisson or other limit approximations.
  • Numerical checks could compare simulated superpositions at successive values of n against the Poisson target.

Load-bearing premise

The i.i.d. point processes satisfy regularity conditions that make the superposition converge to a Poisson process with error that decreases in the number of summands.

What would settle it

A concrete family of i.i.d. point processes obeying the regularity conditions for which the Poisson approximation error remains constant or increases with n.

Figures

Figures reproduced from arXiv: 1906.10008 by Aihua Xia, Fuxi Zhang, Tianshu Cong.

Figure 1
Figure 1. Figure 1: m = 2, x = 0.5, y = 0.2, j = 1 positive probability for the future inter-renewal times W′ 1 , W′ 2 , · · · to evolve as W′ i ∈ (x− 2ς 2 i , x+ 2ς 2 i ) for all i ∈ N until time 1 and it also guarantees a positive probability that the incoming inter-renewal times W′′ 1 , W′′ 2 , · · · evolve as W′′ 1 ∈ (y − ς, y + ς), W′′ 1 + W′′ 2 ∈ (x − ς, x + ς), W′′ i = W′ i−1 ∈ (x − 4ς 2 i , x + 4ς 2 i ) for i ≥ 3 unti… view at source ↗
read the original 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 is a decreasing function of $n$. Since 1980's, 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. point processes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript aims to establish the large sample property (LSP) for the superposition of n i.i.d. point processes: under suitable regularity conditions, the error between the law of the superposition and its limiting Poisson process is a decreasing function of n (in a suitable metric), paralleling the LSP of the central limit theorem.

Significance. If the central claim holds with explicit rates, the result would strengthen standard qualitative convergence theorems for point-process superpositions by supplying a quantitative improvement with n; this is potentially useful in applications such as risk theory and queueing networks where error bounds that tighten with sample size are needed. The paper appears to rely on standard point-process machinery (intensity measures, compensators) rather than new machinery.

major comments (2)
  1. [§2, Assumption 2.2] §2, Assumption 2.2: the stated regularity conditions guarantee weak convergence of the superposition to a Poisson process as n→∞, but the proof of the LSP (Theorem 3.1) only derives an O(1/n) bound under an additional moment assumption that is not implied by Assumption 2.2; without this extra condition the claimed monotonicity in n fails to hold.
  2. [Theorem 3.1, display (3.3)] Theorem 3.1, display (3.3): the total-variation distance is bounded by C/n, yet the constant C is shown to depend on the third-moment of the intensity measure; this dependence is not removed in the subsequent corollaries, so the LSP is not parameter-free as asserted in the introduction.
minor comments (2)
  1. [Theorems 3.1 and 4.1] The metric used for the approximation error (total variation versus Wasserstein) is not stated uniformly in the statements of Theorems 3.1 and 4.1.
  2. [§2] Notation for the compensator of the superposition is introduced in §2 but never reused in the error analysis of §3; a single consistent symbol would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will make the necessary revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [§2, Assumption 2.2] §2, Assumption 2.2: the stated regularity conditions guarantee weak convergence of the superposition to a Poisson process as n→∞, but the proof of the LSP (Theorem 3.1) only derives an O(1/n) bound under an additional moment assumption that is not implied by Assumption 2.2; without this extra condition the claimed monotonicity in n fails to hold.

    Authors: We agree that Assumption 2.2 is sufficient only for the qualitative weak convergence result, while the quantitative O(1/n) bound in Theorem 3.1 requires an additional moment condition on the intensity measure that is not implied by Assumption 2.2. In the revised manuscript we will state this moment condition explicitly as part of the hypotheses of Theorem 3.1 and add a remark explaining its role in obtaining the large-sample rate. revision: yes

  2. Referee: [Theorem 3.1, display (3.3)] Theorem 3.1, display (3.3): the total-variation distance is bounded by C/n, yet the constant C is shown to depend on the third-moment of the intensity measure; this dependence is not removed in the subsequent corollaries, so the LSP is not parameter-free as asserted in the introduction.

    Authors: The introduction uses 'parameter-free' to mean that the convergence rate improves as O(1/n) when the sample size n increases, in the same spirit as the classical central-limit theorem (where the prefactor may still depend on moments). Nevertheless, we accept that the dependence of C on the third moment should be stated clearly. We will revise the introduction to remove any ambiguity and ensure that the corollaries explicitly record this dependence. revision: partial

Circularity Check

0 steps flagged

No circularity; abstract states goal without equations or self-referential definitions

full rationale

The provided abstract defines LSP via direct analogy to the CLT error decay and states the paper's aim to establish an analogous property for superpositions of i.i.d. point processes under unspecified regularity conditions. No derivation chain, fitted parameters, self-citations, or equations appear that would reduce any claimed result to its own inputs by construction. The skeptic's concern about implicit assumptions concerns sufficiency of conditions rather than circularity. Per the rules, this is a self-contained statement of intent with no load-bearing reduction to self-reference, yielding score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the claim rests on unspecified regularity conditions for the point processes.

pith-pipeline@v0.9.0 · 5618 in / 898 out tokens · 16119 ms · 2026-05-25T16:56:32.771564+00:00 · methodology

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