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

Let $({\eta }_i{)}_{i\ge 1}$ be a sequence of $\psi$-mixing random variables. Let $m=\lfloor n^{\alpha }\rfloor ,0<\alpha <1,k=\lfloor n\mathrm{/}(2m)\rfloor ,$ 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 $\mathbb{P}(S_k^o\mathrm{/}\sqrt{[S^o{]}_k}\ge x)$ as $n\to \infty.$ Our result is similar to the recent work of Chen$\mathit{\text{et}}\mathit{\text{al.}}$
[Self-normalized Cram'{e}r-type moderate deviations under dependence. Ann.
Statist.\ 2016; $\mathbf{\text{44}}$(4): 1593–1617] where the authors established Cram'er type moderate deviation expansions for $\beta$-mixing sequences. Comparing to the result of Chen $\mathit{\text{et}}\mathit{\text{al.}}$, our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity.