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Strong laws for 𝐿- and 𝑢-statistics

Jon AaronsonSchool of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, IsraelRobert BurtonDepartment of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605, USAHerold DehlingDepartment of Mathematics, University of Groningen, Groningen, NetherlandsDavid GilatSchool of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, IsraelTheodore P. HillSchool of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USABernard WeissInstitute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

1996en

ABI

Annotatsiya

Strong laws of large numbers are given for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -statistics (linear combinations of order statistics) and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.

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DOI: 10.1090/s0002-9947-96-01681-9

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