Cramér's moderate deviations for martingales with applications

被引:8
|
作者
Fan, Xiequan [1 ,2 ]
Shao, Qi-Man [3 ]
机构
[1] Northeastern Univ Qinhuangdao, Sch Math & Stat, Qinhuangdao 066004, Hebei, Peoples R China
[2] Tianjin Univ, Ctr Appl Math, Tianjin 300072, Peoples R China
[3] Southern Univ Sci & Technol, Natl Ctr Appl Math Shenzhen, Dept Stat & Data Sci, SICM, Shenzhen 518000, Guangdong, Peoples R China
来源
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES | 2024年 / 60卷 / 03期
关键词
Martingales; Cramer's moderate deviations; Berry-Esseen's bounds; Elephant random walks; EXACT CONVERGENCE-RATES; CENTRAL-LIMIT-THEOREM; STATIONARY-SEQUENCES; INEQUALITIES; SUMS;
D O I
10.1214/23-AIHP1372
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
Let (xi i,Fi)i >= 1 i , F i ) i >= 1 be a sequence of martingale differences. Set X n = ni =1 xi i and X n = n i =1 E (xi i 2 |F i -1 ) . We prove Cramer's moderate deviation expansions for P (X n / root X n >= x) and P (X n / E X n 2 >= x) as n ->infinity . Our results extend the classical Cramer result to the cases of normalized martingales Xn/root Xn n / root X n and standardized martingales Xn/ n / E X n 2 , with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.
引用
收藏
页码:2046 / 2074
页数:29
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