Bernstein type inequalities for self-normalized martingales with applications

For self-normalized martingales with conditionally symmetric differences, de la Pe~{n}a [A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, No.1, 537-564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18: 1848-1869] extended de la Pe~{n}a’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 self-normalized sums, t-statistics and autoregressive processes are discussed.