Strong approximation of extended renewal processes
type: Item
instance of: scholarly article
title: Strong approximation of extended renewal processes
zbMATH Open document ID: 0552.60032
author: Lajos Horváth
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: Let \(\{X_ n\}\) be a sequence of i.i.d. random vectors in \(R^ d\), \(d\geq 1\), with expectation \(\mu\) and finite \(\nu\) th moment, \(\nu >2\). Let h be a real function on \(R^ d\) which is homogeneous, \(h(\mu)>0\), and has continuous partial derivatives of the second order in a neighbourhood of \(\mu\). For \(0<q\leq 1\) define the extended renewal process \(N(t)=\min \{k:h(S_ k)\geq tk^{1-q}\}\), \(t\geq 0\), where \(S_ k=X_ 1+...+X_ k.\) The main result of the paper is a strong approximation of \((t/h(\mu))^{1/q}-N(t)\) by a time-transformed Wiener process \(cW((t/h(\mu))^{1/q})\). It is shown that the rate of approximation cannot be improved when \(\nu\geq 4\). Consequences of this theorem are functional and Chung-type laws of iterated logarithm, Bahadur-Kiefer type representation of N(t) in terms of partial sums, and the characterization of N(t) by four classes of deterministic curves in the sense of Révesz. Generalization of the results to the case of nonindependent and nonidentically distributed random vectors is also discussed. It is worthwhile to notice that this paper is the first one considering the case \(q<1\) when \(d>1\).
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 3879820
zbMATH Keywords: renewal process
zbMATH Keywords: strong approximation
zbMATH Keywords: laws of iterated logarithm
zbMATH Keywords: Bahadur-Kiefer type representation
reviewed by: T. Inglot
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993145
OpenAlex ID: W1989473805
DOI: 10.1214/AOP/1176993145
description: scientific article; zbMATH DE number 3879820