Self-normalized Cram\'{e}r type moderate deviations for the maximum of sums

TL;DR

This paper establishes a Cramér type moderate deviation result for the maximum of self-normalized sums of independent zero-mean variables, extending classical results to the maximum and self-normalization context.

Contribution

It introduces a new moderate deviation theorem for the maximum of self-normalized sums, applicable under optimal finite third moment conditions.

Findings

The ratio of the tail probability to the normal tail converges to 2.

Results hold uniformly for x up to o(n^{1/6}).

Applicable to identically distributed variables with finite third moments.

Abstract

Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\sum_{i=1}^nX_i$ and $V^2_n=\sum_{i=1}^nX^2_i$. A Cram\'{e}r type moderate deviation for the maximum of the self-normalized sums $\max_{1\leq k\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\max_{1\leq k\leq n}S_k\geq xV_n)/(1-\Phi (x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$.