Collisions among random walks on a graph
SIAM Journal on Discrete Mathematics
Mixing of random walks and other diffusions on a graph
Surveys in combinatorics, 1995
Efficient stopping rules for Markov chains
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Reversal of Markov Chains and the Forget Time
Combinatorics, Probability and Computing
Random walks and the regeneration time
Journal of Graph Theory
Centers for Random Walks on Trees
SIAM Journal on Discrete Mathematics
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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 $\rX_{\fc{\t}}$ denote the matrix of exit frequencies from singletons to τ* on the reverse chain. We show that $\rX_{\fc{\t}} = R (X_{\t}^{\top}-\vb^{\top} \one)R^{-1}$, 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.