On the limit distribution of discrete random variables
type: Item
instance of: scholarly article
title: On the limit distribution of discrete random variables
zbMATH Open document ID: 0594.60050
DOI: 10.1007/BF00320081
publication date: 1987-01-01T00:00:00Z
review text: It is well-known that the possible limit distributions of triangular arrays \((X_{nk})\) (n\(\geq 1\), \(1\leq k\leq k_ n)\) of infinitesimal random variables with independent rows are exactly the infinitely divisible distributions. In the present paper, it is shown that requiring the random variables \(X_{nk}\) to assume only two values with prescribed probabilities \(p_{nk}\) and \(1-p_{nk}\) does not narrow the class of possible limit distributions, provided that the numbers \(p_{nk}\) satisfy \(0<p_{nk}<1\) and \[ \lim_{n\to \infty}\sum_{1\leq k\leq k_ n, p_{nk}\leq \epsilon}p_{nk}=\infty \quad for\quad every\quad \epsilon >0. \] A similar result is obtained for limit distributions of sequences of independent random variables. As an application to probabilistic number theory, the class of possible limit distributions in the central limit theorem for additive arithmetic functions is determined.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3956128
zbMATH Keywords: triangular arrays
zbMATH Keywords: infinitely divisible distributions
zbMATH Keywords: probabilistic number theory
zbMATH Keywords: central limit theorem
author: Adolf Hildebrand
MaRDI profile type: Publication
cites work: Q3867876
cites work: Q5598785
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3956128