Self-normalized Cram\'er type moderate deviations for stationary sequences and applications

Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^\alpha \rfloor, 0< \alpha < 1,$ and $k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $(V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.