An exponential nonuniform Berry-Esseen bound for self-normalized sums

In this paper we shall derive Exponential nonuniform Berry-Esseen bounds in the central limit theorem for self-normalized sums. We show that the size of the error can be reduced considerably by replacing the usual standardiza tion by self-normalization. Tn particular, we establish the exponential bou nds for the probability of the self-normalized sums under the condition tha t the third moment is finite, whereas an exponential moment assumption is r equired for the standardized sums. Applications to t-statistics and the pro babilities of moderate deviations of self-normalized sums are also discusse d.