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15388 A scaling analysis of a cat and mouse Markov chain
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Litvak, N. and Robert, P. (2009) A scaling analysis of a cat and mouse Markov chain. Memorandum 1899, Department of Applied Mathematics, University of Twente, Enschede. ISSN 1874-4850

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Abstract

Motivated by an original on-line page-ranking algorithm, starting from an arbitrary Markov chain $(C_n)$ on a discrete state space ${\cal S}$, a Markov chain $(C_n,M_n)$ on the product space ${\cal S}^2$, 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 $(C_n)$ 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 $\mathbb{Z}$ and $\mathbb{Z}^2$, reflected simple random walk in $\mathbb{N}$ 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 limit results for occupation times and rare events of Markov processes.

Item Type:Internal Report (Memorandum)
Research Group:EWI-SOR: Stochastic Operations Research
Research Program:CTIT-DSN: Dependable Systems and Networks
Research Project:NetRank: Ranking of Nodes in Complex Stochastic Networks
Additional Information:Paper is deposited at arxiv http://arxiv.org/abs/0905.2259
Uncontrolled Keywords:Cat and mouse Markov chains, Scaling of null recurrent Markov chains, Pagerank algorithms.
ID Code:15388
Deposited On:19 June 2009
More Information:statisticsmetis

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