A weak convergence result for sequential empirical processes under weak dependence

ABSTRACT

The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying α-mixing triangular array of random variables. In particular, the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore, under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as possible applications.

KEYWORDS:

• Sequential empirical processes
• functional central limit theorems
• stochastic equicontinuity
• α-mixing

AMS CLASSIFICATIONS:

• Primary 60F05
• Secondary 60F17

Acknowledgments

I am deeply thankful to my doctoral advisor, Prof. Dr. Natalie Neumeyer, for a careful reading of this manuscript and helpful suggestions. I also want to thank Prof. Dr. Stanislav Volgushev for useful literature suggestions. Furthermore, I sincerely thank the referee and the associate editor for a considerate reading of the paper and helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author.