@TECHREPORT{Kahale_largedeviation, author = {Nabil Kahale}, title = {Large deviation bounds for Markov chains}, institution = {}, year = {} }

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Abstract

We study the fraction of time that a Markov chain spends in a given subset of states. We give an exponential bound on the probability that it exceeds its expectation by a constant factor. Our bound depends on the mixing properties of the chain, and is asymptotically optimal for a certain class of Markov chains. It beats the best previously known results in this direction. We present an application to the leader election problem. 1 Introduction Let (Xm ); m 1, be an irreducible Markov chain on a finite state space V with transition matrix P and stationary distribution ß. We assume that P is reversible, that is ß(u)P (u; v) = ß(v)P (v; u); for all u; v 2 V: (1) Let A be a proper subset of V . Denote by ß(A) = P u2A ß(u) the stationary probability of the set A, and by S l = ØA (X 1 ) + ØA (X 2 ) + \Delta \Delta \Delta + ØA (X l ) the number of steps the Markov chain is inside A. It is known [2] that, for any initial distribution, the fraction S l =l converges almost surely to ß(A) ...