A scaling analysis of a cat and mouse Markov chain

Nelly Litvak, Philippe Robert

Research output: Contribution to journalArticleAcademicpeer-review

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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.
Original languageEnglish
Pages (from-to)792-826
Number of pages30
JournalAnnals of applied probability
Volume22
Issue number2
DOIs
Publication statusPublished - 2012

Fingerprint

Mouse
Markov chain
Scaling
Simple Random Walk
State Space
Occupation Time
Rare Events
Product Space
Invariant Measure
Markov Process
Asymptotic Properties
Null
Continuous Time
Limiting
Asymptotic Behavior

Keywords

  • Scaling of null recurrent Markov chains
  • Cat and mouse Markov chains
  • IR-80034
  • EWI-21025
  • MSC-60J10
  • MSC-90B18
  • METIS-286272

Cite this

Litvak, Nelly ; Robert, Philippe. / A scaling analysis of a cat and mouse Markov chain. In: Annals of applied probability. 2012 ; Vol. 22, No. 2. pp. 792-826.
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A scaling analysis of a cat and mouse Markov chain. / Litvak, Nelly; Robert, Philippe.

In: Annals of applied probability, Vol. 22, No. 2, 2012, p. 792-826.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Robert, Philippe

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AB - 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.

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