Convergence in law of random sums with non-random centering
type: Item
instance of: scholarly article
title: Convergence in law of random sums with non-random centering
zbMATH Open document ID: 0806.60012
DOI: 10.1007/BF02213569
author: Mark Finkelstein
author: V. M. Kruglov
author: Howard G. Tucker
published in: Journal of Theoretical Probability
publication date: 1995-02-14T00:00:00Z
review text: Let \(\{X_ n, n \geq 1\}\) be a sequence of independent random variables such that for some norming constants \(\{B_ n, n \geq 1\}\), \(B_ n \to \infty\), as \(n \to \infty\), and some centering constants \(\{A_ n, n \geq 1\}\) the distribution function of \({1 \over B_ n} \sum^ n_{k = 1} X_ k + A_ n\) converges weakly, as \(n \to \infty\), to some distribution function \(F\). Now suppose \(\{N_ n, n \geq 1\}\) is a sequence of positive integer-valued random variables independent of \(\{X_ n, n \geq 1\}\). One problem of nonrandom centering is this: under what conditions on \(\{N_ n, n \geq 1\}\) does the distribution function of \((X_ 1 + \cdots + X_{N_ n})/B_ n + A_ n\) converge weakly, as \(n \to \infty\), to the same \(F\)? Further, one could also ask under what conditions this sequence converges weakly to any distribution function. Let us observe that the centering by \(A_ n\) is nonrandom centering; random centering would involve \(A_{N_ n}\). Also norming by \(B_ n\) is nonrandom norming; random one would involve \(B_{N_ n}\). The authors present necessary and sufficient conditions for weak convergence of these random variables under nonrandom centering and nonrandom norming, and in some cases, under random norming. The distribution functions for the limit laws are determined, too. Some results concerning necessary and sufficient conditions for weak convergence of random sums of independent random variables have also been obtained by \textit{A. Krajka} and the reviewer [Commentat. Math. Univ. Carol. 34, No. 3, 465-482 (1993; Zbl 0785.60016)].
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 637649
zbMATH Keywords: nonrandom centering
zbMATH Keywords: nonrandom norming
zbMATH Keywords: necessary and sufficient conditions for weak convergence
reviewed by: Zdzisław Rychlik
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 637649