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We prove a new inequality controlling the large deviations of the empiricalmeasure of a Markov chain. This inequality is based on the martingale used byDonsker and Varadhan and the minimax theorem. It holds for convex sets and itrequires to take an infimum over the starting point. In the case of a compactspace, this inequality is a partial improvement of the large deviationsestimates of Donsker and Varadhan. In the case of a non compact space, wecondition on the event that the process visits $n$ times a compact subset ofthe space and we still obtain a control on the exponential scale.

A martingale minimax exponential inequality for Markov chains