Large deviations for martingales via Cramérʹs method

We develop a new approach for proving large deviation results for martingales based on a change of probability measure. It extends to the case of martingales the conjugate distribution technique due to Cramér. To demonstrate our approach, we derive formulae for probabilities of large deviations for martingales with bounded jumps and bounded norming factor. Surprisingly enough, 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.