OPTIMAL HOEFFDING BOUNDS FOR DISCRETE REVERSIBLE MARKOV CHAINS By Carlos

f dπ. From the weak law of large numbers we know that the empirical mean nSn = n −1∑n k=1 f(Xk) converges to μ in probability. This result is the working principle behind all Markov chain Monte Carlo (MCMC) integration techniques. The basis of MCMC dates back to the 50’s with the article of Metropolis, Rosenbluth, Rosenbluth, Teller and Teller (1953), but it is only with today’s computing power that these methods can give their full measure. Like in the classical Monte Carlo schemes, one way of getting insight about the above convergence is by looking at the first moment E[Sn] and the (asymptotic) variance limn −2 V[Sn]. There is abundant literature covering these matters—see, for example, Peskun (1973) and Smith and Roberts (1993). A related problem is also to study the rate at which the chain approaches stationarity. Instead, our concern will be the stationary large deviation probabilities