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In this paper, we study a strong limit theorem of delayed sums for Markov chains in bi-infinite environments by constructing a sequence of random variables with parameters. As a corollary, we obtain some strong limit properties including the generalized conditional relative entropy for Markov chains in bi-infinite random environment. The results which we obtained generalize the results of Liu et al. ( 2015 ).

Some generalized strong limit theorems for Markov chains in bi-infinite random environments