On subordinated distributions and generalized renewal measures
type: Item
instance of: scholarly article
title: On subordinated distributions and generalized renewal measures
zbMATH Open document ID: 0613.60007
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be a sequence of independent, identically distributed random variables with partial sums \(S_ 0=0\), \(S_ n=X_ 1+...+X_ n\). We investigate the behaviour of \(\sum^{\infty}_{n=0}a_ nP(S_ n\in x+A)\) as \(x\to \pm \infty\), where \(a_ 1,a_ 2,..\). is a sequence of nonnegative numbers and \(A\subset {\mathbb{R}}\) is a fixed Borel set. The special case \(a_ n=1\), \(A=(-\infty,0]\), gives the renewal function where our method leads to an expansion down to terms of order \(P(X_ 1\in [x,x+1])\). Other applications are to harmonic renewal measures and infinitely divisible distributions.
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 3992516
zbMATH Keywords: subordinated distributions
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: harmonic renewal measures
zbMATH Keywords: infinitely divisible distributions
author: Rudolf Grübel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992278
OpenAlex ID: W2026924428
DOI: 10.1214/AOP/1176992278
description: scientific article; zbMATH DE number 3992516