Deviation inequalities for martingales with applications

Abstract

Using changes of probability measure developed by Grama and Haeusler (2000) [18], we extend the deviation inequalities of Lanzinger and Stadtmüller (2000) [26] and Fuk and Nagaev (1971) [15] to the case of martingales. Our inequalities recover the best possible decaying rate in the independent case. In particular, these inequalities improve the results of Lesigne and Volný (2001) [27] under a stronger condition that the martingale differences have bounded conditional moments. Applications to linear regressions with martingale difference innovations, weak invariance principles for martingales and self-normalized deviations are provided. In particular, we establish a type of self-normalized deviation bounds for parameter estimation of linear regressions. Such type bounds have the advantage that they do not depend on the distribution of the regression random variables.