Strong laws for 𝐿- and 𝑢-statistics

Abstract

Strong laws of large numbers are given for L L -statistics (linear combinations of order statistics) and for U 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.