Strong Laws for L- and U-Statistics

J. Aaronson, Tel Aviv University
R. Burton, Oregon State University
H. Dehling, Tel Aviv University
D. Gilat, Tel Aviv University
Theodore P. Hill, Georgia Institute of Technology - Main Campus
B. Weiss, Hebrew University of Jerusalem

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This article was first published in Transactions of the American Mathematical Society, published by the American Mathematical Society. Copyright © 1996 American Mathematical Society. The definitive version is available at http://dx.doi.org/10.1090/S0002-9947-96-01681-9.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.