Non-uniform Berry–Esseen bounds for martingales with applications to statistical estimation

ABSTRACT

We establish non-uniform Berry–Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate x's, and are of exponential decay rate as de la Peña's inequality when $x\to \mathrm{\infty }$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.

KEYWORDS:

• Non-uniform Berry–Esseen bound
• Cramér large deviations
• exponential inequality
• linear regressions
• self-normalized large deviations

AMS SUBJECT CLASSIFICATION:

• 60G42
• 60F05
• 60F10
• 60E15
• 62E20
• 62J05

Acknowledgements

The authors are grateful to the reviewers for their comments and remarks which helped to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Fan and Liu have been partially supported by the National Natural Science Foundation of China [Grants no. 11601375, no. 11501146, no. 11571052 and no. 11401590].