Self-normalized Cramér type moderate deviations for the maximum of sums

Abstract

Let $X_{1},X_{2},\ldots$ be independent random variables with zero means and finite variances, and let $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V^{2}_{n}=\sum_{i=1}^{n}X^{2}_{i}$. A Cramér 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 $X_{1},X_{2},\ldots,$ it is proved that $\mathsf{P}(\max_{1\leq k\leq n}S_{k}\geq xV_{n})/(1-\Phi(x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_{1}$.