Bernoulli

  • Bernoulli
  • Volume 19, Number 3 (2013), 1006-1027.

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

Weidong Liu, Qi-Man Shao, and Qiying Wang

Full-text: Open access

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 $X_{1}$.

Article information

Source
Bernoulli, Volume 19, Number 3 (2013), 1006-1027.

Dates
First available in Project Euclid: 26 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1372251151

Digital Object Identifier
doi:10.3150/12-BEJ415

Mathematical Reviews number (MathSciNet)
MR3079304

Zentralblatt MATH identifier
1273.60032

Keywords
independent random variables maximum of self-normalized sums

Citation

Liu, Weidong; Shao, Qi-Man; Wang, Qiying. Self-normalized Cramér type moderate deviations for the maximum of sums. Bernoulli 19 (2013), no. 3, 1006--1027. doi:10.3150/12-BEJ415. https://projecteuclid.org/euclid.bj/1372251151


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