Self-normalized Cramer-type large deviations for independent random variables

Let X-1, X-2,... be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramer-type large deviation result for the standardized partial sums. In this paper, we show that a Cramer-type large deviation theorem holds for self-normalized sums only under a finite (2 + delta)th moment, 0 < delta less than or equal to 1. In particular, we show P(S-n/ V-n greater than or equal to x) = (1 - Phi(x))(1 + O(1)(1 + x)(2+delta)/d(n,delta)(2+delta)) for 0 less than or equal to x less than or equal to d(n,delta), where d(n,delta) = (Sigma(i=1)(n) EXi2)(1/2)/(Sigma(i=1)(n) E\textbackslash{}X-i\textbackslash{}(2+delta))(1/(2+delta)) and V-n = (Sigma(i=1)(n) X-i(2))(1/2). Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed