A scaling analysis of a cat and mouse Markov chain
type: Item
instance of: scholarly article
title: A scaling analysis of a cat and mouse Markov chain
zbMATH Open document ID: 1253.60081
DOI: 10.1214/11-AAP785
author: Nelly Litvak
author: Philippe Robert
published in: The Annals of Applied Probability
publication date: 2012-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.2259
full work available at URL: https://projecteuclid.org/euclid.aoap/1333372013
reviewed by: Oleg K. Zakusilo
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 90B18
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 6034170
zbMATH Keywords: cat and mouse Markov chains
zbMATH Keywords: scaling of null recurrent Markov chains
MaRDI profile type: Publication
arXiv ID: 0905.2259
cites work: Applied Probability and Queues
cites work: A Survey on PageRank Computing
cites work: Exponential functionals of Lévy processes
cites work: Q4269108
cites work: Limit theorems for occupation times of Markov processes
cites work: Q3928760
cites work: Q4714465
cites work: Random walks on weighted graphs and applications to on-line algorithms
cites work: A discrete renewal theorem with infinite mean
cites work: Q5342183
cites work: Implicit renewal theory and tails of solutions of random equations
cites work: Q4003764
cites work: AIMD algorithms and exponential functionals
cites work: A limit theorem for slowly increasing occupation times
cites work: A limit theorem for sums of random number of i.i.d. random variables and its application to occupation times of Markov chains
cites work: Markov chain models - rarity and exponentiality
cites work: Q4118602
cites work: Q5595646
cites work: Random Walks and A Sojourn Density Process of Brownian Motion
cites work: Markov Chains
cites work: Weak invariance principles for local time
cites work: Q4411306
cites work: Q4508926
cites work: Q4328337
cites work: Q3128901
cites work: Exponential functionals of Brownian motion and related processes
OpenAlex ID: W3104724492
review text: The paper considers a Markov chain \(({C_n},{M_n})\) on the discrete state space \((S \times S)\). The sequence \(({C_n})\), representing the location of the cat, is a Markov chain with transition matrix \(P = (p(x,y))\) associated to the random walk on the graph \(S\). The second coordinate, the location of the mouse, \(({M_n})\) has the following dynamics: (a) if \({M_n} \neq {C_n}\), then \({M_{n + 1}} = {M_n}\); (b) if \({M_n} = {C_n}\), then, depending on \({M_n}\), the random variable \({M_{n + 1}}\) has distribution \(p({M_n},y)\) and is independent of \({C_{n + 1}}\).NEWLINENEWLINEThe asymptotic properties of \(({C_n},{M_n})\) for a number of transition matrices \(P\) are the subject of this paper. A representation of its invariant measure is, in particular, obtained. When the state space is infinite it is shown that this Markov chain is in fact null recurrent if the initial Markov chain \(({C_n})\) is positive recurrent and reversible. In this context, the scaling properties of the location of the second component are investigated in various situations.
description: scientific article; zbMATH DE number 6034170