  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.
