Self-normalized Cramér type moderate deviations for martingales and applications

被引：0

作者：

Fan, Xiequan ^{[1
]}

Shao, Qi-man ^{[2
]}

机构：

[1] Northeastern Univ Qinhuangdao, Sch Math & Stat, Qinhuangdao, Peoples R China

[2] Southern Univ Sci & Technol, Natl Ctr Appl Math Shenzhen, Dept Stat & Data Sci, SICM, Shenzhen 518000, Peoples R China

来源：

BERNOULLI | 2025年 / 31卷 / 01期

基金：

中国国家自然科学基金;

关键词：

Branching process in a random environment; Cram & eacute; r's moderate deviations; martingales; self-normalized sequences; Student's statistic; SUBCRITICAL BRANCHING-PROCESSES; RANDOM ENVIRONMENT; LIMIT-THEOREMS; STATIONARY-SEQUENCES; MOMENTS;

D O I：

10.3150/24-BEJ1722

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：

Cram & eacute;r's moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimators used in statistics. In this paper, we establish self-normalized Cram & eacute;r type moderate deviations for martingales under some mild conditions. The result extends an earlier work of Fan et al. (Bernoulli 25 (2019) 2793-2823). Moreover, applications of our result to Student's statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed.

引用

页码：130 / 161

页数：32

共 45 条

[31] Novak S.Y., 2012, Monographs on Statistics and Applied Probability, V122
[32] A maximal Lp-inequality for stationary sequences and its applications
Peligrad, Magda
Utev, Sergey
Wu, Wei Biao
[J]. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2007, 135 (02) : 541 - 550
[33] Petrov V. V., 1975, Sums of independent random variables, DOI [10.1007/978-3-642-65809-9, DOI 10.1007/978-3-642-65809-9]
[34] LARGE DEVIATIONS FOR MARTINGALES WITH SOME APPLICATIONS
RACKAUSKAS, A
[J]. ACTA APPLICANDAE MATHEMATICAE, 1995, 38 (01) : 109 - 129
[35] Rackauskas A., 1990, Liet. Mat. Rink., V30, P784, DOI [10.1007/BF00970833, DOI 10.1007/BF00970833]
[36] Saulis L., 1978, Limit Theorems for Large Deviations
[37] Cramer type moderate deviation theorems for self-normalized processes
Shao, Qi-Man
Zhou, Wen-Xin
[J]. BERNOULLI, 2016, 22 (04) : 2029 - 2079
[38] Self-normalized limit theorems: A survey
Shao, Qi-Man
Wang, Qiying
[J]. PROBABILITY SURVEYS, 2013, 10 : 69 - 93
[39] A Cramer type large deviation result for student's t-statistic
Shao, QM
[J]. JOURNAL OF THEORETICAL PROBABILITY, 1999, 12 (02) : 385 - 398
[40] Shao QM, 1997, ANN PROBAB, V25, P285

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