Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 2 , pp. 195--200

AN INVARIANCE PRINCIPLE FOR $U$-STATISTICS WITH THE DEGREE OF THE KERNEL DEPENDING UPON THE SAMPLE SIZE

By

CHANDRAKANT M. DEO, *University of Ottawa*

SUMMARY. We prove an invariance principle for $U$-statistics where the degree of the kernel is allowed to depend upon the sample size. As an immediate consequence, we obtain a functional from of a recent theorem by van Es and Helmers (1988).

*AMS (1990) subject classification.* 60F,62G.

*Key words and phrases.* Invariance principles, $U$-statistics, symmetric statistics.