Total variation distance for Poisson subset numbers
type: Item
instance of: scholarly article
title: Total variation distance for Poisson subset numbers
zbMATH Open document ID: 1106.60008
author: Larry Goldstein
author: Gesine Reinert
published in: Annals of Combinatorics
publication date: 2007-01-05T00:00:00Z
review text: Let \(A_0,A_1,\dots,A_k\) be subsets of \(\{1,2,\dots,n\}\) with fixed sizes \(a_0,a_1,\dots,a_k\) taken at random independently and uniformly from the all subsets with given sizes and let \(W=| \bigcap_{j=0}^k| \) be the size of their intersection. Explicit total variation bound is given between the distributions of \(W\) and the Poisson distribution with intensity \(\lambda_{k,n}=EW.\) The obtained bound allows the authors to show that the distribution of \(W\) converges to a Poisson distribution if \(a_i\to\infty\), \(i=0,1,\dots,k\), in such a way that \(\lambda_{k,n}\lambda_k\in (0,\infty).\) Reviewer's note: Considering \(\bar{A}_i\) instead of \(A_i\), \(i=0,1,\dots,k\) and treating the numbers \(n-a_0,n-a_1,\dots,n-a_k\) as the sizes of groups of particles, we arrive to the well-known problem of equiprobable allocation of the \(k+1\) groups particles of sizes \(n-a_0,n-a_1,\dots,n-a_k\) into \(n\) cells in such a way that each cell contains at most one particle of each group. In this case \(W\) is the number of empty cells after the allocation of all groups of particles. Under such interpretation Poisson and normal limit theorems for \(W\) (including estimates of the total variation distance) were obtained by the reviewer and \textit{V. G. Mikhailov} [Theory Probab. Appl. 27, 734--743 (1982); translation from Teor. Veroyatn. Primen. 27, No. 4, 684--692 (1982; Zbl 0517.60008)] in slightly different terms.
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 62E17
zbMATH DE Number: 5082145
zbMATH Keywords: Poisson approximation
zbMATH Keywords: Stein's method
zbMATH Keywords: size biasing
zbMATH Keywords: surprisology
Wikidata QID: Q57747987
reviewed by: Vladimir Vatutin
MaRDI profile type: Publication
full work available at URL: s00026-006-0291-9
OpenAlex ID: W2095534553
DOI: 10.1007/S00026-006-0291-9
Recommended article: Poisson limit theorems in allocation schemes of distinguishable particles
Recommended article: Limit theorems for the number of empty cells
Recommended article: The Poisson limit distribution of the number of empty cells in the generalized equiprobable scheme of allocating particles by groups
Recommended article: Q4272122
Recommended article: Poisson limit theorems for number of given value cells in non-homogeneous generalized allocation scheme
Recommended article: Poisson approximation for random sums of Bernoulli random variables
Recommended article: On asymptotic expansion in the random allocation of particles by sets
Recommended article: On the asymptotic normality of the number of empty cells in a scheme of group allocation of particles
Recommended article: Rates of Poisson convergence for some coverage and urn problems using coupling
description: scientific article; zbMATH DE number 5082145