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

Abstract

Let (Xi)i≥1 be a stationary sequence. Denote m=⌊nα⌋,0<α<1, and k=⌊n∕m⌋, where ⌊a⌋ stands for the integer part of a. Set Sj∘=∑i=1mXm(j−1)+i,1≤j≤k, and (Vk∘)2=∑j=1k(Sj∘)2. We prove a Cramér type moderate deviation expansion for P(∑j=1kSj∘∕Vk∘≥x) as n→∞. Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.