### Abstract

Original language | English |
---|---|

Pages (from-to) | 792-826 |

Number of pages | 30 |

Journal | Annals of applied probability |

Volume | 22 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 |

### Fingerprint

### Keywords

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

### Cite this

*Annals of applied probability*,

*22*(2), 792-826. https://doi.org/10.1214/11-AAP785

}

*Annals of applied probability*, vol. 22, no. 2, pp. 792-826. https://doi.org/10.1214/11-AAP785

**A scaling analysis of a cat and mouse Markov chain.** / Litvak, Nelly; Robert, Philippe.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - A scaling analysis of a cat and mouse Markov chain

AU - Litvak, Nelly

AU - Robert, Philippe

N1 - eemcs-eprint-21025

PY - 2012

Y1 - 2012

N2 - 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.

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.

KW - Scaling of null recurrent Markov chains

KW - Cat and mouse Markov chains

KW - IR-80034

KW - EWI-21025

KW - MSC-60J10

KW - MSC-90B18

KW - METIS-286272

U2 - 10.1214/11-AAP785

DO - 10.1214/11-AAP785

M3 - Article

VL - 22

SP - 792

EP - 826

JO - Annals of applied probability

JF - Annals of applied probability

SN - 1050-5164

IS - 2

ER -