A General Class of Exponential Inequalities for Martingales and Ratios

Abstract

In this paper we introduce a technique for obtaining exponential inequalities, with particular emphasis placed on results involving ratios. Our main applications consist of approximations to the tail probability of the ratio of a martingale over its conditional variance (or its quadratic variation for continuous martingales). We provide examples that strictly extend several of the classical exponential inequalities for sums of independent random variables and martingales. The spirit of this application is that, when going from results for sums of independent random variables to martingales, one should replace the variance by the conditional variance and the exponential of a function of the variance by the expectation of the exponential of the same function of the conditional variance. The decoupling inequalities used to attain our goal are of independent interest. They include a new exponential decoupling inequality with constraints and a sharp inequality for the probability of the intersection of a fixed number of dependent sets. Finally, we also present an exponential inequality that does not require any integrability conditions involving the ratio of the sum of conditionally symmetric variables to its sum of squares.