Limit theorems and structural properties of the cat-and-mouse markov chain and its generalisations

Sergey Foss, Timofei Prasolov, Seva Shneer

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Abstract

We revisit the so-called cat-and-mouse Markov chain, studied earlier by Litvak and Robert (2012). This is a two-dimensional Markov chain on the lattice Z2, where the first component (the cat) is a simple random walk and the second component (the mouse) changes when the components meet. We obtain new results for two generalisations of the model. First, in the two-dimensional case we consider far more general jump distributions for the components and obtain a scaling limit for the second component. When we let the first component be a simple random walk again, we further generalise the jump distribution of the second component. Secondly, we consider chains of three and more dimensions, where we investigate structural properties of the model and find a limiting law for the last component.
Original languageEnglish
Pages (from-to)141-166
Number of pages26
JournalAdvances in Applied Probability
Volume54
Issue number1
Early online date28 Feb 2022
DOIs
Publication statusPublished - Mar 2022

Keywords

  • Cat-and-mouse games
  • compound renewal process
  • multidimensional Markov chain
  • randomly stopped sums
  • regular variation
  • weak convergence

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics

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