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

Let $X_1,X_2,\dots$ be i.i.d. random variables with zero mean and variance ${\sigma }^2\in (0,\mathrm{\infty })$. 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, $2<p\le 3$. In particular, we show that $\mathbb{P}(M_n\ge x{V}_n)=2(1-\mathrm{\Phi }(x))\{1+O(1){\beta }_{n,p}(1+x{)}^p\}$ for $0\le x\le {\beta }_{n,p}^{-1\mathrm{/}p}$, where $M_n={\max }_{1\le k\le n}{\sum }_{j=1}^k{X}_j$, $V_n=({\sum }_{j=1}^n{X}_j^2{)}^{1\mathrm{/}2}$, ${\beta }_{n,p}=\mathbb{E}\mathrm{\mid }{X}_1{\mathrm{\mid }}^p\mathrm{/}(n^{(p-2)\mathrm{/}2}{\sigma }^p)$, and $\mathrm{\Phi }$ is the standard normal distribution function. This proves a conjecture of Shao2010. More precisely, we bound the relative error by the same truncated-moment quantity as for self-normalized sums. The proof compares the maximum with the absolute sum while preserving their common denominator, through a reflected-walk representation and Fourier estimates with distribution-adapted smoothing.