The Annals of Statistics

Self-normalized Cramér-type moderate deviations under dependence

Xiaohong Chen, Qi-Man Shao, Wei Biao Wu, and Lihu Xu

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We establish a Cramér-type moderate deviation result for self-normalized sums of weakly dependent random variables, where the moment requirement is much weaker than the non-self-normalized counterpart. The range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes. We consider three types of self-normalization: the equal-block scheme, the big-block-small-block scheme and the interlacing scheme. Simulation study shows that the latter can have a better finite-sample performance. Our result is applied to multiple testing and construction of simultaneous confidence intervals for ultra-high dimensional time series mean vectors.

Article information

Ann. Statist. Volume 44, Number 4 (2016), 1593-1617.

Received: April 2015
Revised: December 2015
First available in Project Euclid: 7 July 2016

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Mathematical Reviews number (MathSciNet)

Primary: 62E20: Asymptotic distribution theory 60F10: Large deviations

Cramér-type moderate deviation absolutely regular functional dependence measures ultra-high dimensional time series


Chen, Xiaohong; Shao, Qi-Man; Wu, Wei Biao; Xu, Lihu. Self-normalized Cramér-type moderate deviations under dependence. Ann. Statist. 44 (2016), no. 4, 1593--1617. doi:10.1214/15-AOS1429.

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Supplemental materials

  • Supplement to “Self-normalized Cramér type moderate deviations under dependence”. The supplement gives the detailed proof for Theorem 4.2.