Moderate deviations for martingales and mixing random processes

We obtain a moderately large deviation theorem for martingales. Then this result is applied to prove that the empirical measures of a stationary ∅-mixing sequence of random variables satisfy moderately large deviation principle when Σ+∞n=1 ∅(n) < + ∞. Another application shows that the empirical measures of a Markov process obey uniformly moderately large deviation principle under Doeblin recurrence.