A note on Poisson approximation in multivariate case
type: Item
instance of: scholarly article
title: A note on Poisson approximation in multivariate case
zbMATH Open document ID: 0633.60034
author: Cheng-Gee Liu
published in: Kodai Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: In recent years has been discussed the Poisson approximation for the sum of independent Bernoulli random vectors. We usually define multivariate Poisson distributions P(\(\lambda)\) by \(P(x=K)=\sum_{[C]}\prod_{i\in E}p(\alpha_ i;\lambda_ i)\) where \(p(\alpha_ i;\lambda_ i)\) is a univariate Poisson density. \textit{K. Kawamura} [ibid. 2, 337-345 (1979; Zbl 0434.60019)] has derived sufficient conditions of a Poisson approximation to the sum of independent identically multivariate Bernoulli random vectors. The author has discussed the multivariate Poisson distribution by the limiting value of the sum of Bernoulli random vectors and shown sufficient conditions for the Poisson approximation to the sum of independent Bernoulli random vectors which may or may not be identically distributed. The converse assertion of this paper will also be true which needs very complicated proofs and will be published in the near future, that is, the conditions are also necessary for the Poisson approximation. Finally, this paper gives a trivial result if we suppose that the sequence of probability distributions has the property of ``smallness'', that is, \(\lim_{k}\min_{j}P(X_{kj}=0)=1\).
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4030607
zbMATH Keywords: Poisson approximation for the sum of independent Bernoulli random vectors
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037417
OpenAlex ID: W2121565456
cites work: Poisson approximation for sums of independent bivariate Bernoulli vectors
cites work: Necessary and sufficient conditions for a Poisson approximation (trivariate case)
cites work: The structure of multivariate Poisson distribution
cites work: A Family of Bivariate Distributions Generated by the Bivariate Bernoulli Distribution
DOI: 10.2996/KMJ/1138037417
description: scientific article; zbMATH DE number 4030607