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For an irreducible Markov chain $(X_n)_{n\ge 0}$ we identify the rate function governing the large deviation estimation of empirical mean $\frac {1}{n} \sum_{k=0}^{n-1} f(X_k)$ by means of the Donsker-Varadhan's entropy. That allows us to obtain the lower bound of large deviations for the empirical measure $\frac {1}{n} \sum_{k=0}^{n-1} \delta_{X_k}$ in full generality

Identification of the rate function for large deviations of an irreducible Markov chain