A note on Poisson approximation of rescaled set-indexed empirical processes
type: Item
instance of: scholarly article
title: A note on Poisson approximation of rescaled set-indexed empirical processes
zbMATH Open document ID: 0952.60015
DOI: 10.1016/S0167-7152(99)00094-2
published in: Statistics \& Probability Letters
publication date: 2001-01-04T00:00:00Z
review text: The author studies Poisson approximation to the multivariate empirical process in a small neighbourhood of 0. Given i.i.d. random variables \(\{X_i,i\geq 1\}\) with values in \(\mathbb{R}^k\) define the point process \(S_n(A)= \sum_{i\leq n}I (n^{1/k} X_i\in A)\), \(A\in{\mathcal B}_0\), where \({\mathcal B}_0\) is the class of all Borel subsets of some bounded set \(A_0\subset \mathbb{R}^k\). If \(X_i\) has a density \(f\) which is continuous at 0, the process \(S_n\) can be defined on the same probability space as a Poisson process \(\Pi\) with intensity \(f(0)\cdot \lambda\) in such a way that \(P(\sup_{A\in{\mathcal B}_0}|S_n(A)-\Pi (A)|> 0)\to 0.\) The proof uses Le Cam's result on Poisson approximation in separable normed spaces to get a bound on the total variation distance between \(S_n\) and \(\Pi\) and Dohrushin's coupling lemma.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 1395086
zbMATH Keywords: Poisson approximation
zbMATH Keywords: multivariate empirical process
zbMATH Keywords: total variation distance
author: Igor S. Borisov
reviewed by: Herold Dehling
MaRDI profile type: Publication
cites work: Poisson approximation of the partial sum process in Banach spaces
cites work: Prescribing a System of Random Variables by Conditional Distributions
cites work: Poisson convergence for set-indexed empirical processes
cites work: Q5639057
description: scientific article; zbMATH DE number 1395086