Statistica Sinica **11**(2001), 289-302

Charles Heilig and Deborah Nolan

Abstract:In this paper, we provide uniform limit theory for a U-statistic of increasing degree, also called an infinite-degree U-statistic. The stochastic process based on collections of U-statistics is referred to as a U-process, and if the U-statistic is infinite-degree, we have an infinite-degree U-process. Frees (1986) proposed a nonparametric renewal estimator which is an infinite-degree U-statistic. In a later paper, Frees (1989) provided conditions for the pointwise asymptotic theory for the infinite-degree U-statistic. To extend the pointwise results to limit theory for the infinite-degree U-process that holds uniformly over the index set, we build on existing results for U-processes of fixed degree. In particular we extend the symmetrization techniques of Nolan and Pollard (1987) and the moment inequalities of Sherman (1994) to obtain uniform weak laws of large numbers and functional central limit theory for the infinite-degree U-process.

Key words and phrases:Empirical process, functional central limit theory, U-statistic, uniform weak law of large numbers.