Large deviations for martingales

Let (X-i) be a martingale difference sequence and S-n = Sigma (n)(i=1) X-i. We prove that if sup(i)E(e(\ Xi \)) < infinity then there exists c > 0 such that mu (S-n > n) less than or equal to e(-cn1/3); this bound is optimal for the class of martingale difference sequences which are also strictly stationary and ergodic. If the sequence (X-i) is bounded in L-p, 2 less than or equal to p < infinity, then we get the estimation mu (S-n > n) less than or equal to 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, mu (S-n > n) = o(n(1-p)) for X-i is an element of L-p, 1 less than or equal to p < infinity, cannot be improved and that, reciprocally, it implies the integrability of \X-i\ (p-delta) for all delta > 0. (C) 2001 Elsevier Science B.V. All rights reserved.