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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Strong laws for $L$- and $u$-statistics
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by J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill and B. Weiss PDF
Trans. Amer. Math. Soc. 348 (1996), 2845-2866 Request permission

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.
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Additional Information
  • J. Aaronson
  • Affiliation: School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
  • Email: aaro@math.tau.ac.il
  • R. Burton
  • Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605, USA
  • Email: burton@math.orst.edu
  • H. Dehling
  • Affiliation: Department of Mathematics, University of Groningen, Groningen, Netherlands
  • Email: dehling@math.rug.nl
  • D. Gilat
  • Affiliation: School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
  • Email: gilat@math.tau.ac.il
  • T. Hill
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USA
  • Email: hill@math.gatech.edu
  • B. Weiss
  • Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
  • MR Author ID: 181570
  • Email: weiss@math.huji.ac.il
  • Received by editor(s): March 31, 1995
  • Additional Notes: Burton was partially supported by AFOSR grant 91-0215, and NSF grant DMS-9163738. Gilat and Hill were partially supported by U.S./Israel Binational Science Foundation grant 88-00005. Hill was also supported by NSF grant DMS-9209586, and Dutch National Science Foundation (NWO) dossier B-61-281.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 2845-2866
  • MSC (1991): Primary 60F15, 62G05; Secondary 28D99, 62G30
  • DOI: https://doi.org/10.1090/S0002-9947-96-01681-9
  • MathSciNet review: 1363941