Some improved results on Berry–Esséen bounds for strong mixing random variables and applications

Abstract

In this paper, we mainly establish a general form of Berry–Esséen bound for α-mixing random variables. With different choices of the parameters, the rates are shown as $O(n^{-3/26})$, $O(n^{-1/6})$, and approximately $O(n^{-1/4})$. These results improved some corresponding ones in the literature. An application to the Berry–Esséen bound of sample quantiles is further provided. Moreover, some simulations are also carried out to support the theoretical results.

Keywords:

• Berry–Esséen bound
• α-mixing random variables
• sample quantiles
• central limit theorem

Mathematical Subject Classifications:

• 62F12
• 62E20
• 60F05

Acknowledgments

The authors are most grateful to the Editor and three anonymous reviewers for carefully reading the manuscript and for valuable suggestions which helped in improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supported by the National Social Science Foundation of China [grant number 22BTJ059].