Self-normalized Cramér type moderate deviations for stationary sequences and applications

Abstract

Let $(X_i{)}_{i\ge 1}$ be a stationary sequence. Denote $m=\lfloor n^{\alpha }\rfloor ,0<\alpha <1,$ and $k=\lfloor n/m\rfloor ,$ where $\lfloor a\rfloor$ stands for the integer part of $a.$ Set $S_j^{\circ }={\sum }_{i=1}^m{X}_{m(j-1)+i},1\le j\le k,$ and $(V_k^{\circ }{)}^2={\sum }_{j=1}^k(S_j^{\circ }{)}^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}({\sum }_{j=1}^k{S}_j^{\circ }/V_k^{\circ }\ge x)$ as $n\to \infty .$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.

30 pages

References in corpus (2)

• Exponential inequalities for self-normalized martingales with applications
• A Cramér moderate deviation theorem for Hotelling's $T^2$-statistic with applications to global tests

Cited by in corpus (3)

• Self-normalized Cramér type moderate deviations for martingales and applications
• Self-normalized Cramér-type Moderate Deviation of Stochastic Gradient Langevin Dynamics
• Central limit theorem and Self-normalized Cramér-type moderate deviation for Euler-Maruyama Scheme