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Limit theorems for the infinite-degree $$U$$-process. (English) Zbl 1057.62518
Summary: 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 of 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.

##### MSC:
 62G99 Nonparametric inference 60F17 Functional limit theorems; invariance principles 62E20 Asymptotic distribution theory in statistics