Large deviations for martingales via Cramer's method

We develop a new approach for proving large deviation results for martingal es based on a change of probability measure. It extends to the case of mart ingales the conjugate distribution technique due to Cramer. To demonstrate our approach, we derive formulae for probabilities of large deviations for martingales with bounded jumps and bounded norming factor. Surprisingly eno ugh, our result shows that the relative error in the normal range is of the same order as in the case of sums of independent random variables. It also allows to extend the range beyond the normal one. (C) 2000 Elsevier Scienc e B.V. All rights reserved.