August 2024 Cramér’s moderate deviations for martingales with applications
Xiequan Fan, Qi-Man Shao
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(3): 2046-2074 (August 2024). DOI: 10.1214/23-AIHP1372

Abstract

Let (ξi,Fi)i1 be a sequence of martingale differences. Set Xn=i=1nξi and Xn=i=1nE(ξi2|Fi1). We prove Cramér’s moderate deviation expansions for P(Xn/Xnx) and P(Xn/EXn2x) as n. Our results extend the classical Cramér result to the cases of normalized martingales Xn/Xn and standardized martingales Xn/EXn2, with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.

Soit (ξi,Fi)i1 une suite de différences de martingale. Soient Xn=i=1nξi et Xn=i=1nE(ξi2|Fi1). Nous prouvons les développements de déviation modérée de Cramér pour P(Xn/Xnx) et P(Xn/EXn2x) lorsque n. Nos résultats étendent le résultat classique de Cramér aux cas des martingales normalisées Xn/Xn et des martingales standardisées Xn/EXn2, où les différences de martingale vérifient la condition de Bernstein conditionnelle. Des applications aux marches aléatoires des éléphants et aux processus autorégressifs sont également discutées.

Funding Statement

Fan X. was partially supported by the National Natural Science Foundation of China (Grant no. 11971063). Shao Q.M. was partially supported by the National Natural Science Foundation of China (Grant no. 12031005) and Shenzhen Outstanding Talents Training Fund.

Acknowledgements

Fan is deeply indebted to Prof. Ion Grama and Prof. Quansheng Liu for introducing his to the study of martingale limit theory.

Citation

Download Citation

Xiequan Fan. Qi-Man Shao. "Cramér’s moderate deviations for martingales with applications." Ann. Inst. H. Poincaré Probab. Statist. 60 (3) 2046 - 2074, August 2024. https://doi.org/10.1214/23-AIHP1372

Information

Received: 15 March 2022; Revised: 26 October 2022; Accepted: 23 January 2023; Published: August 2024
First available in Project Euclid: 31 July 2024

Digital Object Identifier: 10.1214/23-AIHP1372

Subjects:
Primary: 60E20 , 60F10 , 60G42
Secondary: 62M10

Keywords: Berry–Esseen’s bounds , Cramér’s moderate deviations , Elephant random walks , Martingales

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré

Vol.60 • No. 3 • August 2024
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