Bernstein type inequalities for self-normalized martingales with applications

For self-normalized martingales with conditionally symmetric differences, de la Pena [A general class of exponential inequalities for martingales and ratios. Ann Probab. 1999;27(1):537-564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann Appl Probab. 2008;18:1848-1869] extended de la Pena's inequalities to martingales with differences heavy on left. In this paper, we establish Bernstein type exponential inequalities for self-normalized martingales with differences bounded from below. Moreover, applications to t-statistics and autoregressive processes are discussed.