### Abstract

Consider a finite irreducible Markov chain on state space S with transition matrix M and stationary distribution π. Let R_{ii} = 1/π_{i}abe the diagonal matrix of return times, Rii = 1/i. Given distributions σ, τ and k ∈ S, the exit frequency x _{k}(σ τ) denotes the expected number of times a random walk exits state k before an optimal stopping rule from to halts the walk. For a target distribution τ, we define X_{t} as the n × n matrix given by (X_{τ})_{ij} = x_{j}(i,τ), where i also denotes the singleton distribution on state i. The dual Markov chain with transition matrix = Ř M^{τ}R^{-1} is called the reverse chain. We prove that Markov chain duality extends to matrices of exit frequencies. Specifically, for each target distribution τ^{*}, we associate a unique dual distribution τ^{*}. Let X̌_{τ}* denote the matrix of exit frequencies from singletons to τ_{*} on the reverse chain. We show that X̌_{τ}* = R (X^{τ}_{τ}-b ^{τ}1)R^{-1}, where b is a non-negative constant vector (depending on τ ). We explore this exit frequency duality and further illuminate the relationship between stopping rules on the original chain and reverse chain.

Original language | English |
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Pages (from-to) | 541-560 |

Number of pages | 20 |

Journal | Combinatorics Probability and Computing |

Volume | 19 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 1 2010 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Combinatorics Probability and Computing*,

*19*(4), 541-560. https://doi.org/10.1017/S0963548310000118