SELF-NORMALIZED CRAMER TYPE MODERATE DEVIATION THEOREM FOR GAUSSIAN APPROXIMATION

• 成果类型: Article
• 署名作者: Qiu, Jingkun; Chen, Song Xi; Shao, Qi-Man
• 署名单位: Peking University; Tsinghua University; Southern University of Science & Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/25-AOS2507
• 发表日期: 2025
• 页码: 1319-1346
• 关键词: CENTRAL-LIMIT-THEOREM; high-dimensional means; Change-point Detection; bootstrap approximations; 2-sample test; inference; regression; sums; time; INDEPENDENCE

摘要

Berry-Esseen type bounds for Gaussian approximation of standardized sums have been extensively studied under exponential type moment conditions. In this paper, a Cramer type moderate deviation theorem is established for self-normalized Gaussian approximation under finite moment conditions. More specifically, let X-1, X-2, ...,X-n be i.i.d. R-p-valued random vectors with zero means. Let Sn,j =Sigma(n)(i=1) Xij and V-2 (n,j) = Sigma(n)(i=1) X-2 (ij) . We show that if the correlation matrix of X-1 is I-p and the third moment of X1 is finite, then P(max(1 <= j <= p)S(n,j)/V-n,V-j > x)/P(max(1 <= j <= p) Z(j) > x) -> 1 uniformly for 0 <= x <= o(n(1/6)) and for all p >= 1, where Z(1),... ,Z(p) are independent standard normal random variables. Similar result is also established for large x when X-1 has a general correlation matrix. The proof is based on a new Cramer type moderate deviation theorem for the minimum of several self-normalized sums. As an application, we propose a high dimensional one-sample t-test that allows for an exponential growth of p without requiring the commonly used sub-Gaussian assumption.