Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums

Let $X_1,X_2,\ldots$ be i.i.d. random variables with zero mean and variance $σ^2\in(0,\infty)$. In this paper, we establish a sharp Cramér-type moderate deviation for the self-normalized maximum of partial sums under a finite $p$-th absolute moment, $2<p\le3$. In particular, we show that $\mathbb{P}