Deviation inequalities for martingales with applications to linear regressions and weak invariance principles

TL;DR

This paper extends deviation inequalities for martingales using probability measure changes, achieving optimal decay rates and applying these results to linear regressions and weak invariance principles.

Contribution

It generalizes existing deviation inequalities for martingales, improving their applicability and accuracy in dependent data scenarios.

Findings

Derived new deviation inequalities for martingales.

Achieved optimal decay rates matching independent cases.

Applied inequalities to linear regression models and invariance principles.

Abstract

Using changes of probability measure developed by \mbox{Grama} and Haeusler (Stochastic Process.\ Appl., 2000), we obtain two generalizations of the deviation inequalities of Lanzinger and Stadtm\"{u}ller (Stochastic Process.\ Appl., 2000) and Fuk and Nagaev (Theory Probab. Appl., 1971) to the case of martingales. Our inequalities recover the best possible decaying rate of independent case. Applications to linear regressions and weak invariance principles for martingales are provided.