A renewal theorem of Blackwell type
type: Item
instance of: scholarly article
title: A renewal theorem of Blackwell type
zbMATH Open document ID: 0537.60087
author: Paul Embrechts
author: M. Maejima
author: Edward Omey
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: Let \(S_ n\) be the sum of the first n terms of a sequence of i.i.d. random variables with distribution F having positive finite mean \(\mu\), and let \(G(x)=\sum^{\infty}_{n=1}a(n)P(S_ n\leq x)\), where a(x) is a positive function which varies regularly at infinity with exponent \(\alpha\). It is shown that if \(\alpha>-1\) and F is non-lattice then \(G(x+h)-G(x)\sim h\mu^{-\alpha -1}a(x)\), for \(h>0\), as \(x\to \infty\). This asymptotic relation holds also for \(\alpha\leq -1\) under some additional assumptions.
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 3854162
zbMATH Keywords: generalized renewal measures
zbMATH Keywords: regular variation
zbMATH Keywords: Blackwell theorem
reviewed by: Toshio Mori
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993305
OpenAlex ID: W2034184055
DOI: 10.1214/AOP/1176993305
description: scientific article; zbMATH DE number 3854162