An Exponential Nonuniform Berry-Esseen Bound for Self-Normalized Sums

Abstract

In this paper we shall derive exponential nonuniformBerry–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 standardization by self-normalization. In particular, we establish the exponential bounds for the probability of the self-normalized sums under the condition that the third moment is finite, whereas an exponential moment assumption is required for the standardized sums. Applications to t-statistics and the probabilities of moderate deviations of self-normalized sums are also discussed.