Estimation of the limit variance for sums under a new weak dependence condition

ABSTRACT

We prove a self-normalized central limit theorem for a mixing class of processes introduced in Kacem M, Loisel S, Maume-Deschamps V. [Some mixing properties of conditionally independent processes. Commun Statist Theory Methods. 2016;45:1241–1259]. This class is larger than more classical strongly mixing processes and thus our result is more general than [Peligrad M, Shao QM. Estimation of the variance of partial sums for ρ-mixing random variables. J Multivar Anal. 1995;52:140–157; Shi S. Estimation of the variance for strongly mixing sequences. Appl Math J Chinese Univ. 2000;15(1):45–54] ones. The fact that some conditionally independent processes satisfy this kind of mixing properties motivated our study. We investigate the weak consistency as well as the asymptotic normality of the estimator of the variance that we propose.

KEYWORDS:

• Self normalized central limit theorem
• mixing processes
• invariance principle
• variance estimation

Acknowledgements

We are grateful to anonymous referees and to the Associate Editor whose valuable suggestions and remark allowed us to greatly improve the original version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.