Optimal Chernoff and Hoeffding Bounds for Finite Markov Chains
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This paper develops an optimal Chernoff type bound for the probabilities of large deviations of sums ∑_k=1^n f (X_k) where f is a real-valued function and (X_k)_k ∈N_0 is a finite Markov chain with an arbitrary initial distribution and an irreducible stochastic matrix coming from a large class of stochastic matrices. Our bound is optimal in the large deviations sense attaining a constant prefactor and an exponential decay with the optimal large deviations rate. Moreover through a Pinsker type inequality and a Hoeffding type lemma, we are able to loosen up our Chernoff type bound to a Hoeffding type bound and reveal the sub-Gaussian nature of the sums. Finally we show a uniform multiplicative ergodic theorem for our class of Markov chains.
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