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Abstract. Let {X"; n > 0} be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times {N¡; i > 1} such that {XN.; i > 1} are independent and identically distributed. This idea is used to show that {Xn} is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.

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Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.