Asymptotics of maxima of discrete random variables
type: Item
instance of: scholarly article
title: Asymptotics of maxima of discrete random variables
zbMATH Open document ID: 1035.60032
DOI: 10.1023/A:1024081112501
author: K. V. Mitov
author: Saralees Nadarajah
published in: Extremes
publication date: 2004-03-16T00:00:00Z
review text: Let \(X\) be a discrete random variable on the nonnegative integers. Let \(X_{1},\ldots ,X_{n}\) be independent and identically distributed copies of \(X\), and let \(M_{n}=\max(X_{1},\ldots,X_{n})\). The authors study the limiting distribution of \((M_{n}-b(n))/a(n)\) as \(n\to\infty\) for some real numbers \(a(n)>0\) and \(b(n)\). There are derived non-degenerate limit laws for uniform, binomial, geometric and negative binomial distributions (by letting their parameters vary as \(n\to\infty\)). For example the following theorem is proved. Let \(X_{1},\dots ,X_{n}\) be i.i.d. uniform random variables with parameter \(N=N(n)\). If \(N(n)\) grows with \(n\) according to \(n=o(N(n))\), then there are sequences \(a(n)=N(n)/n\), \(b(n)=N(n)\) such that \(\text{Pr}\{M_{n}\leq a(n)x+b(n)\}\to \exp(x)\) as \(n\to\infty\).
reviewed by: Aleksandr D. Borisenko
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 2059071
zbMATH Keywords: asymptotics
zbMATH Keywords: maxima of discrete random variables
zbMATH Keywords: binomial
zbMATH Keywords: geometric
zbMATH Keywords: negative binomial
zbMATH Keywords: Poisson
zbMATH Keywords: uniform
MaRDI profile type: Publication
full work available at URL: a:1024081112501
OpenAlex ID: W1734814728
description: scientific article; zbMATH DE number 2059071