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

Abstract

Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n={\sum }_{i=1}^n{X}_i$ and $V_n^2={\sum }_{i=1}^n{X}_i^2$. A Cramér type moderate deviation for the maximum of the self-normalized sums ${\max }_{1\le k\le n}{S}_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P({\max }_{1\le k\le n}{S}_k\ge x{V}_n)/(1-\Phi (x))\to 2$ uniformly for $0<x\le \mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$.

Published in at http://dx.doi.org/10.3150/12-BEJ415 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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