Generalized Poisson distributions as limits of sums for arrays of dependent random vectors
type: Item
instance of: scholarly article
title: Generalized Poisson distributions as limits of sums for arrays of dependent random vectors
zbMATH Open document ID: 0821.60032
author: Maria Kobus
published in: Journal of Multivariate Analysis
publication date: 1995-09-27T00:00:00Z
review text: Let \(\{X_{k,n}, k \in Z, n \in N\}\), where \(Z\) is the set of integers and \(N\) that of natural numbers, be arrays of random vectors in \(R^ d\) and let \((k_ n)\) be a sequence of integers with \(k_ n \to \infty\) as \(n \to \infty\). Suppose that the sequence \(\{X_{k,n}, k \in Z, n \in N\}\) is stationary in rows and set \(S_ n = \sum^ n_{k = 1} X_{k,n}\), \(n \in N\). The author obtains necessary and sufficient conditions for \((S_ n)\), properly normalized, to converge to a generalized Poisson distribution, when \(\{X_{k,n}, k \in Z\}\) is an \(m\)-dependent or mixing \((\alpha\)-, \(\rho\)-, \(\varphi\)-mixing) sequence. For \((X_ n)\) a strictly stationary sequence of random vectors, under some mixing conditions, the author obtains a functional limit theorem for the partial sum process. Point process theory is a major tool used here. This paper extends some of the results by \textit{A. Jakubowski} and the author [ibid. 29, No. 2, 219-251 (1989; Zbl 0687.60025)].
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 739443
zbMATH Keywords: limit theorems for sums
zbMATH Keywords: dependent sequences
zbMATH Keywords: generalized Poisson processes
zbMATH Keywords: point process theory
zbMATH Keywords: strictly stationary sequence
zbMATH Keywords: functional limit theorem
MaRDI profile type: Publication
full work available at URL: jmva.1995.1011
OpenAlex ID: W2059806371
DOI: 10.1006/JMVA.1995.1011
description: scientific article; zbMATH DE number 739443