Cramér type moderate deviations for self-normalized $\psi$-mixing sequences

Abstract

Let $({\eta }_i{)}_{i\ge 1}$ be a sequence of $\psi$-mixing random variables. Let $m=\lfloor n^{\alpha }\rfloor ,0<\alpha <1,k=\lfloor n/(2m)\rfloor ,$ and $Y_j={\sum }_{i=1}^m{\eta }_{m(j-1)+i},1\le j\le k.$ Set $S_k^o={\sum }_{j=1}^k{Y}_j$ and $[S^o{]}_k={\sum }_{i=1}^k(Y_j{)}^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}(S_k^o/\sqrt{[S^o{]}_k}\ge x)$ as $n\to \infty .$ Our result is similar to the recent work of Chen\textit{ et al.}\ [Self-normalized Cramér-type moderate deviations under dependence. Ann.\ Statist.\ 2016; \textbf{44}(4): 1593--1617] where the authors established Cramér type moderate deviation expansions for $\beta$-mixing sequences. Comparing to the result of Chen \textit{et al.}, our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity.

18 pages

References in corpus (3)

• Exponential inequalities for self-normalized martingales with applications
• Cramér large deviation expansions for martingales under Bernstein's condition
• Cramér's moderate deviations for martingales with applications