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arxiv: 2604.23928 · v1 · submitted 2026-04-27 · 🧮 math.ST · math.PR· stat.TH

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Wasserstein convergence rates for empirical measures of point processes

Dongzhou Huang , Tianyi Jiang , Haonan Wang
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Pith reviewed 2026-05-07 17:57 UTC · model grok-4.3

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keywords pointprocessesmeasuresconvergenceempiricalresultswassersteindistance
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The pith

Sharp convergence rates and concentration bounds are established for empirical measures of point processes under a newly introduced Wasserstein distance on counting measures.

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

Point processes are random collections of points in space or time, such as earthquake locations or customer arrivals. Researchers often want to know how well a sample of observed points represents the underlying pattern. The paper defines a new way to measure distance between two such collections of points by first creating a metric on counting measures and then using that to build a Wasserstein-type distance. With this distance they prove both upper and lower bounds on how quickly the empirical distribution from a finite sample approaches the true distribution. They also obtain concentration results that quantify how likely the sample is to be close to the truth. These bounds are described as sharp, meaning they cannot be improved in general. The results are positioned as tools for statistical tests and inference on point process data.

Core claim

we establish sharp upper and lower bounds on the convergence rate of the empirical measures of point processes under the Wasserstein distance

Load-bearing premise

The new metric on the space of counting measures is well-defined and induces a Wasserstein distance that makes the convergence statements hold for the point processes under study (details of the metric construction and any regularity conditions on the processes are not visible in the abstract).

Figures

Figures reproduced from arXiv: 2604.23928 by Dongzhou Huang, Haonan Wang, Tianyi Jiang.

Figure 1
Figure 1. Figure 1: Left Panel: The dashed and solid lines are the cumulative function for view at source ↗
read the original abstract

In this paper, we establish sharp upper and lower bounds on the convergence rate of the empirical measures of point processes under the Wasserstein distance. To this end, we first introduce a new metric on the space of counting measures and, based on this metric, define a Wasserstein distance between point processes. We then employ it to study the convergence rate of the empirical measures of point processes, which serves as a natural tool for identifying the distribution of the underlying point process. Furthermore, we derive concentration results. These theoretical results provide constructive tools for hypothesis testing and statistical inference for point processes. The applicability of our results is demonstrated through several practical examples.

Editorial analysis

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Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

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pith-pipeline@v0.9.0 · 5400 in / 1037 out tokens · 31229 ms · 2026-05-07T17:57:47.340922+00:00 · methodology

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Works this paper leans on

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