Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified

Abstract

We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v = delta mu t ], is at most exp(-Omega( delta^2 mu t / T )) for 0 1. In fact, the bounds hold even if the weight functions f_i's for i in [t] are distinct, provided that all of them have the same mean mu. We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on the spectral expansion lambda of M, which is the square root of the second largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t| >= delta mu t ] is at most exp(-Omega( delta^2 (1-lambda) mu t )) for 0 1. Both of our results extend to continuous-time Markov chains, and to the case where the walk starts from an arbitrary distribution x, at a price of a multiplicative factor depending on the distribution x in the concentration bounds.