Moderate deviations principles for self-normalized martingales

We prove, for martingales self-normalized by their increasing process, the upper bound of a moderate deviations principle. Self-normalizing allows to get rid of the of exponential convergence of the previsible square variation which appears in previous works on a deterministic normalization of the martingale. The proof relies on the notion of partial large deviations principle introduced by Dembo and Shao in [3] and [4]. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.