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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Strong laws for $L$- and $U$-Statistics


Authors: J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill and B. Weiss
Journal: Trans. Amer. Math. Soc. 348 (1996), 2845-2866
MSC (1991): Primary 60F15, 62G05; Secondary 28D99, 62G30
MathSciNet review: 1363941
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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
Email: weiss@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01681-9
PII: S 0002-9947(96)01681-9
Keywords: $L$-statistic, $L$-estimator, $L$-parameter, $U$-statistic, $U$-parameter, $V$-statistic, strong law of large numbers, ergodic stationary process
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.
Article copyright: © Copyright 1996 American Mathematical Society