Unbiased time-average estimators for Markov chains
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We consider a time-average estimator f_k of a functional of a Markov chain. Under a coupling assumption, we show that the expectation of f_k has a limit μ as the number of time-steps goes to infinity. We describe a modification of f_k that yields an unbiased estimator f̂_k of μ. It is shown that f̂_k is square-integrable and has finite expected running time. Under certain conditions, f̂_k can be built without any precomputations, and is asymptotically at least as efficient as f_k, up to a multiplicative constant arbitrarily close to 1. Our approach provides an unbiased estimator for the bias of f_k. We study applications to volatility forecasting, queues, and the simulation of high-dimensional Gaussian vectors. Our numerical experiments are consistent with our theoretical findings.
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