LIMIT THEOREMS FOR THE INFINITE-DEGREE U-PROCESS

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.

[1]  Allan Gut,et al.  On the Moments and Limit Distributions of Some First Passage Times , 1974 .

[2]  Robert P. Sherman,et al.  Maximal Inequalities for Degenerate $U$-Processes with Applications to Optimization Estimators , 1994 .

[3]  Joseph P. Romano,et al.  Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions , 1994 .

[4]  G. Pisier.  Some applications of the metric entropy condition to harmonic analysis , 1983 .

[5]  Weak convergence of infinite order U-processes , 1991 .

[6]  W. Hoeffding.  A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[7]  Edward W. Frees.  Nonparametric Renewal Function Estimation , 1986 .

[8]  Guninar Blom.  Some properties of incomplete U-statistics , 1976 .

[9]  A. Bonami.  Étude des coefficients de Fourier des fonctions de $L^p(G)$ , 1970 .

[10]  R. Grübel,et al.  Nonparametric Estimation in Renewal Theory I: The Empirical Renewal Function , 1993 .

[11]  Alan J. Lee,et al.  U-Statistics: Theory and Practice , 1990 .

[12]  R. Dudley.  A course on empirical processes , 1984 .

[13]  E. Giné,et al.  Limit Theorems for $U$-Processes , 1993 .

[14]  Edward W. Frees,et al.  Infinite Order U-statistics , 1989 .

[15]  Deborah Nolan,et al.  Functional Limit Theorems for $U$-Processes , 1988 .

[16]  D. Pollard,et al.  $U$-Processes: Rates of Convergence , 1987 .

[17]  Helmut Schneider,et al.  Comparison of Nonparametric Estimators for the Renewal Function , 1990 .

[18]  Grace S. Shieh.  Infinite order V-statistics , 1994 .

[19]  A. Mandelbaum,et al.  Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals , 1983 .

[20]  V. Peña.  Decoupling and Khintchine's Inequalities for $U$-Statistics , 1992 .

[21]  C. O'Cinneide,et al.  Asymptotics of the Sample Renewal Function , 1995 .