On large deviation probabilities for self-normalized sums of random variables

TL;DR

This paper develops a new approach to analyze large deviation probabilities for self-normalized sums, extending classical results and providing exact asymptotics with a graphical interpretation, applicable to multivariate cases.

Contribution

It introduces a reduction technique linking self-normalized sums to bivariate random walks, generalizes classical theorems, and offers exact large deviation asymptotics with graphical insights.

Findings

Proved classical large deviation theorem under more general conditions

Provided exact asymptotics for large deviation probabilities

Extended results to multivariate self-normalized sums

Abstract

We reduced the large deviation problem for a self-normalized random walk to one for an auxiliary usual bivariate random walk. This enabled us to prove the classical theorem for self-normalized walks by Q.-M. Shao (1997) under slightly more general conditions and, moreover, to provide a graphical interpretation for the emerging limit in terms of the rate function for the bivariate problem. Furthermore, using this approach, we obtained exact (rather than just logarithmic) large deviation asymptotics for the probabilities of interest. Extensions to more general self-normalizing setups including the multivariate case were discussed.