Exact asymptotic for tail of distribution of self-normalized

TL;DR

This paper provides precise asymptotic formulas and estimates for the tail distribution of self-normalized sums of random variables, including non-standard norming, without requiring independence or identical distribution, under smoothness conditions.

Contribution

It introduces exact asymptotic results and non-asymptotic bounds for the tail behavior of self-normalized sums with general conditions, extending previous work to broader settings.

Findings

Derived asymptotic tail behavior for self-normalized sums.

Established non-asymptotic estimates under smoothness conditions.

Demonstrated the sharpness of conditions with counterexamples.

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).