On recurrence and transience of self-interacting random walks
type: Item
instance of: scholarly article
title: On recurrence and transience of self-interacting random walks
zbMATH Open document ID: 1301.60060
author: Yuval Peres
author: S. Popov
author: Perla Sousi
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2014-06-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.3459
review text: Take \(\mu_1,\mu_2,\dots{},\mu_k\) probability measures on \(\mathbb R^d\) with zero mean and supports containing \(d\)-linearly independent vectors of \(\mathbb R^d\). Consider the following (possibly non-Markovian) random walk \((X_n)_{n\in\mathbb N}\): at each time \(i\in\mathbb N\), choose one of the \(k\) measures according to a strategy that depends only on the trajectory \((X_0,\dots{},X_i)\) of the walk up to time \(i\) and perform the next step of the walk according to the chosen measure. The authors deal with the problem of transience VS recurrence for this kind of walks. The main result of the paper states that, if \(k=2\), \(d\geq 3\) and \(\mu_1,\mu_2\) have finite \(2+\beta\) moments, with \(\beta>0\), then \((X_n)_{n\in\mathbb N}\) is transient for any possible choice of the strategy. On the other hand, for \(k=d\), the authors exhibit explicit examples of measures and strategies that produce walks that are recurrent. Note that these two results together give, in particular, a complete picture of the case \(d=3\).
reviewed by: Michele Salvi
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60K37
zbMATH DE Number: 6307358
zbMATH Keywords: random walk
zbMATH Keywords: transience
zbMATH Keywords: recurrence
zbMATH Keywords: Lyapunov function
MaRDI profile type: Publication
OpenAlex ID: W1980708177
arXiv ID: 1203.3459
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DOI: 10.1007/S00574-013-0036-4
description: scientific article