Strong law of large numbers for a class of U-statistics /
This dissertation studies the strong law of large numbers for U-statistics with kernel h : Rm-- R, h(xl,... , xm) = x1.....xm, and normalizing sequence - yn. satisfying some regularity conditions, in several instances, namely, when the indices of summation are restricted to a square, to a strip or t...
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| Format: | Thesis Book |
| Language: | English |
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[Place of publication not identified] :
[publisher not identified] ;
1996.
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| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=739363491&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | This dissertation studies the strong law of large numbers for U-statistics with kernel h : Rm-- R, h(xl,... , xm) = x1.....xm, and normalizing sequence - yn. satisfying some regularity conditions, in several instances, namely, when the indices of summation are restricted to a square, to a strip or to a sector in Zm+. For m = 2, results analogous to the ones in Cuzick, Gine and Zinn (1995) are obtained for 2-sample U- statistics, and under some regularity assumptions on the distributions of the random variables, these results are generalized to m > 2. Convergence in a strip of Z2+ is characterized for both one-sample and 2-sample U- statistics, and it is shown to be equivalent to convergence in the square. Convergence in a sector of Zm of the normalized maxima of products is proved to be equivalent to convergence in the square. Similar results are obtained for one-sample and 2-sample U- statistics in a sector Z2+. Finally, a complete characterization of those powers of order statistics for which there exists a constant c e R+, such that whenever Xi, i = 1, . . . , k are independent and identically distributed non-negative random variables, and X(1) <- (2) <... < Xlk) are the order statistics, we have EXr1......Xrk<=cEX91) is given. |
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| Item Description: | Vita. "Major Subject: Mathematics". |
| Physical Description: | v,57 leaves ; 28 cm. Issued also on microfiche from University Microfilms Inc. |
| Bibliography: | Includes bibliographical references: pages 55-56. |