Large deviations for martingales

Let (Xi) be a martingale difference sequence and Sn=∑i=1n Xi. We prove that if supi E(e|Xi|)<∞ then there exists c>0 such that μ(Sn>n)⩽e−cn1/3; this bound is optimal for the class of martingale difference sequences which are also strictly stationary and ergodic. If the sequence (Xi) is bounded in Lp, 2⩽p<∞, then we get the estimation μ(Sn>n)⩽cn−p/2 which is again optimal for strictly stationary and ergodic sequences of martingale differences. These estimations can be extended to martingale difference fields. The results are also compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, μ(Sn>n)=o(n1−p) for Xi∈Lp, 1⩽p<∞, cannot be improved and that, reciprocally, it implies the integrability of |Xi|p−δ for all δ>0.