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Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and $U$-statistics

Part of: Limit theorems Distribution theory Stochastic processes

Published online by Cambridge University Press:  30 September 2022

Federico Pianoforte  and
Riccardo Turin Open the ORCID record for Riccardo Turin [Opens in a new window]
Federico Pianoforte*
Affiliation:
University of Bern
Riccardo Turin*
Affiliation:
University of Bern
*
*Postal address: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland.
*Postal address: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland.
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Abstract

This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of the Stein equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_\pi$ , stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure, and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in $d_\pi$ are as good as those already available in the literature.

Keywords

Chen–Stein methodWasserstein distancemultinomial distributionm-dependent random vectorsPapangelou intensity

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes 62E17: Approximations to distributions (nonasymptotic)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 223 - 240
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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