Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums
Abstract
Let X1,X2,… be i.i.d. random variables with zero mean and variance σ2∈(0,∞). 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≤3. In particular, we show that ℙ(Mn≥ xVn)=2(1−Φ(x)){1+O(1)βn,p(1+x)p} for 0≤ x≤βn,p−1/p, where Mn=max1≤ k≤ n∑j=1kXj, Vn=(∑j=1nXj2)1/2, βn,p=𝔼|X1|p/(n(p−2)/2σp), 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.
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BibTeX
@misc{liu2026sharp,
title = {{Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums}},
author = {Song-Hao Liu and Qi-Man Shao and Hao Shi and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21975},
eprint = {2610.21975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21975},
}
RIS
TY - UNPB
TI - Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Hao Shi
AU - Jing-Yu Xu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21975
UR - https://arxiv.org/abs/2610.21975
AB - Let X_1,X_2,… be i.i.d. random variables with zero mean and variance σ^2∈(0,∞). 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≤3. In particular, we show that ℙ(M_n≥ xV_n)=2(1−Φ(x)){1+O(1)β_n,p(1+x)^p} for 0≤ x≤β_n,p^−1/p, where M_n=max_1≤ k≤ n∑_j=1^kX_j, V_n=(∑_j=1^nX_j^2)^1/2, β_n,p=𝔼|X_1|^p/(n^(p−2)/2σ^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.
ER -