The Annals of Applied Probability

A scaling analysis of a cat and mouse Markov chain

Nelly Litvak and Philippe Robert

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Abstract

If (Cn) is a Markov chain on a discrete state space $\mathcal{S}$, a Markov chain (Cn, Mn) on the product space $\mathcal{S}\times\mathcal{S}$, the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov chain are investigated. 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 (Cn) is positive recurrent and reversible. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in ℤ and ℤ2 reflected a simple random walk in ℕ and also in a continuous time setting. For several of these processes, a time scaling with rapid growth gives an interesting asymptotic behavior related to limiting results for occupation times and rare events of Markov processes.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 2 (2012), 792-826.

Dates
First available in Project Euclid: 2 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1333372013

Digital Object Identifier
doi:10.1214/11-AAP785

Mathematical Reviews number (MathSciNet)
MR2953569

Zentralblatt MATH identifier
1253.60081

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90B18: Communication networks [See also 68M10, 94A05]

Keywords
Cat and mouse Markov chains scaling of null recurrent Markov chains

Citation

Litvak, Nelly; Robert, Philippe. A scaling analysis of a cat and mouse Markov chain. Ann. Appl. Probab. 22 (2012), no. 2, 792--826. doi:10.1214/11-AAP785. https://projecteuclid.org/euclid.aoap/1333372013


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