Exact asymptotic for tail of distribution of self-normalized

Abstract

We derive in this article the asymptotic behavior as well as non-asymptotical estimates of tail of distribution for self-normalized sums of random variables (r.v.) under natural classical norming. We investigate also the case of non-standard random norming function and the tail asymptotic for the maximum distribution for self-normalized statistics. We do not suppose the independence or identical distributionness of considered random variables, but we assume the existence and sufficient smoothness of its density. We show also the exactness of our conditions imposed on the considered random variables by means of building of an appropriate examples (counterexamples).