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Exit Frequency Matrices for Finite Markov Chains

Published online by Cambridge University Press:  14 May 2010

ANDREW BEVERIDGE  and
LÁSZLÓ LOVÁSZ
ANDREW BEVERIDGE
Affiliation:
Department of Mathematics and Computer Science, Macalester College, Saint Paul, MN 55105, USA (e-mail: abeverid@macalester.edu)
LÁSZLÓ LOVÁSZ
Affiliation:
Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary (e-mail: lovasz@cs.elte.hu)
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Abstract

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Consider a finite irreducible Markov chain on state space S with transition matrix M and stationary distribution π. Let R be the diagonal matrix of return times, Rii = 1/πi. Given distributions σ, τ and k ∈ S, the exit frequency xk(σ, τ) denotes the expected number of times a random walk exits state k before an optimal stopping rule from σ to τ halts the walk. For a target distribution τ, we define Xτ as the n × n matrix given by (Xτ)ij = xj(i, τ), where i also denotes the singleton distribution on state i.

The dual Markov chain with transition matrix = RM⊤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 denote the matrix of exit frequencies from singletons to τ* on the reverse chain. We show that , 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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 4 , July 2010 , pp. 541 - 560
Copyright
Copyright © Cambridge University Press 2010

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