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Strong laws for \(L\)- and \(U\)-statistics. (English) Zbl 0863.60032

Summary: 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 i.i.d. 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.

MSC:

60F15 Strong limit theorems
62G05 Nonparametric estimation
28D99 Measure-theoretic ergodic theory
62G30 Order statistics; empirical distribution functions
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