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

Source abstract

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 $\mathbb{P}(M_n\ge xV_n)=2(1-Φ(x))\{1+O(1)β_{n,p}(1+x)^p\}$ for $0\le x\leβ_{n,p}^{-1/p}$, where $M_n=\max_{1\le k\le n}\sum_{j=1}^kX_j$, $V_n=(\sum_{j=1}^nX_j^2)^{1/2}$, $β_{n,p}=\mathbb{E}|X_1|^p/(n^{(p-2)/2}σ^p)$, and $Φ$ 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.