Bilateral tail estimate for distribution of self normalizes sums of independent centered random variables under natural norming

TL;DR

This paper provides exact non-asymptotic exponential and power estimates for self-normalized sums of independent centered random variables using Grand Lebesgue Spaces, advancing understanding of their distributional properties.

Contribution

It introduces precise non-asymptotic bounds for self-normalized sums leveraging Grand Lebesgue Spaces, a novel approach in this context.

Findings

Derived exact exponential estimates for self-normalized sums.

Established power estimates under natural norming.

Applied Grand Lebesgue Spaces theory to obtain these bounds.

Abstract

We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of random variables.