Continuous time vertex-reinforced jump processes
type: Item
instance of: scholarly article
title: Continuous time vertex-reinforced jump processes
zbMATH Open document ID: 1009.60027
author: Burgess Davis
author: Stanislav Volkov
publication date: 2002-12-01T00:00:00Z
review text: The aim of this article is the study of a vertex-reinforced jump process conceived by W. Werner. This is a continuous time right-continuous nearest neighbour random walk \(X\) on the integers which jumps to a (nearest) neighbour at the rate of one plus the total time spent at that neighbour previously. The authors show that the portion of time before \(t\) the process spends at some integer \(i\) converges for \(t\rightarrow\infty\) to a positive random variable \(V_i\). The sum of \(V_i\) over the integers is one and their joint distribution is explicitly described. Moreover, they prove that \(\lim_{t\rightarrow\infty}\max_{0\leq s\leq t}X_s/\log t\approx 2.77\). The symmetry of the law of \(X\) gives the analog for the minimum. As a consequence \(X\) is recurrent. The study of vertex-reinforced jump processes is motivated by the studies on bond-reinforced random walks by Coppersmith and Diaconis and on vertex-reinforced random walks by \textit{R. Pemantle} [Ann. Probab. 16, No. 3, 1229-1241 (1988; Zbl 0648.60077)] and \textit{R. Pemantle} and \textit{S. Volkov} [ibid. 27, No. 3, 1368-1388 (1999; Zbl 0960.60041)].
reviewed by: Andreas Martin
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 1836958
zbMATH Keywords: reinforced process
zbMATH Keywords: pure birth process
zbMATH Keywords: Pólya urn
MaRDI profile type: Publication
OpenAlex ID: W2110315389
DOI: 10.1007/S004400100189
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1836958