Cramér type moderate deviations for self-normalized $ψ$-mixing sequences

Abstract

Let $(η_i)_{i\geq1}$ be a sequence of $ψ$-mixing random variables. Let $m=\lfloor n^α\rfloor, 0< α< 1, k=\lfloor n/(2m) \rfloor,$ and $Y_j = \sum_{i=1}^m η_{m(j-1)+i}, 1\leq j \leq k.$ 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 $\mathbb{P}(S_k^o/\sqrt{[ S^o]_k} \geq x)$ 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 $β$-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.