A scaling analysis of a cat and mouse Markov chain (Q417084): Difference between revisions

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.