Deviation inequalities, moderate deviation principles for certain Gaussian functionals, and their applications in parameter estimation

ABSTRACT

In this article, we study the deviation inequalities, moderate deviation principle (MDP) and Berry–Esseen bounds for certain Gaussian functionals arising from the Ornstein-Uhlenbeck process without tears. As an application, several asymptotic properties for the minimum distance estimator are obtained. The main methods include the MDP  and deviation inequality for multiple Wiener–Itô integrals.

KEYWORDS:

• Deviation inequalities
• minimum distance estimator
• moderate deviation principle
• multiple Wiener–Itô integrals
• Ornstein–Uhlenbeck process

MATHEMATICS SUBJECT CLASSIFICATION:

• 62N02
• 60F15
• 60G50

Acknowledgments

The authors would like to express gratitude to the anonymous referee for the careful reading and constructive comments which led to an improved presentation of this article.

Funding

Hui Jiang is supported by the National Natural Science Foundation of China under grant (No. 11101210), the Fundamental Research Funds for the Central Universities under grant (NS2015074), and China Postdoctoral Science Foundation under grant (No. 2013M531341, No. 2016T90450). Shimin Li is supported by Natural Science Foundation of Jiangsu Province, China (No. BK20160788). Shaochen Wang is supported by China Postdoctoral Science Foundation under grant (No. 2015M580713) and National Natural Science Foundation of China (No. 11571262, No. 11271140).