Stopping rule reversal for finite Markov chains

Consider a finite irreducible Markov chain with transition matrix M = (pij). Fixing a target distribution τ , we study a family of optimal stopping rules from the singleton distributions to τ . We show that this family of rules is dual to a family of (not necessarily optimal) rules on the reverse chain from the singleton distributions to a related distribution α̂ called the τ -contrast distribution. This duality can be expressed using matrices of exit frequencies. Furthermore, we identify a third distribution β called the τ -core such that α̂ and β are entirely dual to one another: a family of optimal rules to α̂ on the reverse chain are dual to a family of optimal rules to β on the forward chain. Using this duality, we provide new proofs of some exact mixing measure results of Lovász and Winkler [11] concerning the mixing time, the reset time and the forget time. In addition, we show that the time between independent samples of α̂ on the reverse chain is equal to the time between independent samples of β on the forward chain. Finally, we study the properties of matrices of exit frequencies for optimal families of stopping rules. We show that the inverse of an exit frequency matrix can be obtained via an alteration of the singular matrix I − M . Our observations lead to new proofs of two spectral results for mixing measures.