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

Abstract

Let (ηi)i≥1 be a sequence of ψ-mixing random variables. Let m=⌊nα⌋,0<α<1,k=⌊n/(2m)⌋, and Yj=∑i=1mηm(j−1)+i,1≤j≤k. Set Sko=∑j=1kYj and [So]k=∑i=1k(Yj)2. We prove a Cramér type moderate deviation expansion for P(Sko/[So]k≥x) as n→∞. Our result is similar to the recent work of Chen et al. (2016) [4] where the authors established Cramér type moderate deviation expansions for β-mixing sequences. Comparing to the result of Chen et al., our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity.