Théorie du renouvellement pour des chaînes semi-markoviennes transientes. (Renewal theory for transient semi-Markov chains)
type: Item
instance of: scholarly article
title: Théorie du renouvellement pour des chaînes semi-markoviennes transientes. (Renewal theory for transient semi-Markov chains)
zbMATH Open document ID: 0681.60095
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77337
review text: A Markov chain \(((X_ n,Y_ n))\) with state space \(X\times {\mathbb{R}}^ d\) is termed semi-Markovian if the transition kernel P satisfies \[ \int P((x,y+z),dx'\times dy')f(x',y')=\int P((x,y),dx'\times dy')f(x',y'+z) \] for each x,y and z. The author studies the renewal theory of such processes, showing, in particular, that under appropriate ergodicity hypotheses on the Markov chain \((X_ n)\) (and other assumptions), the renewal kernel (potential kernel) \(U=\sum^{\infty}_{n=1}P^ n\) satisfies \[ \lim_{| y| \to \infty}\sqrt{\det \sigma}| y|_{\sigma}^{d-1}U f(x,y)=c_ d\int F\quad (w,y')\pi (dw)dy' \] for each function f and each x, where \(\sigma\) is a certain positive- definite matrix, \[ | y|_{\sigma}=\sqrt{\sigma^{-1}(y)},\quad c_ d=\Gamma ((d-2)/2)/(2\pi)^{d/2}, \] and \(\pi\) is the invariant distribution of \((X_ n)\). This result generalizes the classical renewal theorems of Blackwell for ordinary renewal processes on \({\mathbb{R}}\) and Ney and Spitzer for random walks on \({\mathbb{R}}^ d\).
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 4115700
zbMATH Keywords: semi-Markov process
zbMATH Keywords: renewal theory
zbMATH Keywords: ergodicity hypotheses
zbMATH Keywords: renewal theorems of Blackwell
author: Martine Babillot
reviewed by: Alan F. Karr
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4115700