Cram\'{e}r type moderate deviations for self-normalized $\psi$-mixing sequences

TL;DR

This paper establishes Cramér type moderate deviation results for self-normalized sums of $ ext{psi}$-mixing sequences, extending previous work to sequences with polynomial decay of mixing coefficients and broader applicability.

Contribution

It provides the first Cramér type moderate deviation expansion for self-normalized sums of $ ext{psi}$-mixing sequences with polynomial decay, broadening the scope of dependence structures covered.

Findings

Results hold for polynomial decay of mixing coefficients.

Extends moderate deviation results to wider dependence ranges.

Similar to recent work but for $ ext{psi}$-mixing sequences.

Abstract

Let $(\eta_i)_{i\geq1}$ be a sequence of $\psi$-mixing random variables. Let $m=\lfloor n^\alpha \rfloor, 0< \alpha < 1, k=\lfloor n/(2m) \rfloor,$ and $Y_j = \sum_{i=1}^m \eta_{m(j-1)+i}, 1\leq j \leq k.$ Set $ S_k^o=\sum_{j=1}^{k } Y_j $ and $[S^o]_k=\sum_{i=1}^{k } (Y_j )^2.$ We prove a Cram\'er 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 Cram\'{e}r-type moderate deviations under dependence. Ann.\ Statist.\ 2016; \textbf{44}(4): 1593--1617] where the authors established Cram\'er 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.