Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-02-10T08:10:20.429Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-parameter strong laws and maximal inequalities for U-statistics*

Published online by Cambridge University Press:  14 November 2011

Terry R. McConnell
Terry R. McConnell
Affiliation:
Department of Mathematics, 200 Carnegie, Syracuse University, Syracuse NY 13244-1150, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 107 , Issue 1-2 , 1987 , pp. 133 - 151
Copyright
Copyright © Royal Society of Edinburgh 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aldous, D.. Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 (1981), 581–598.CrossRefGoogle Scholar
2Burkholder, D. L.. Maximal inequalities as necessary conditions for almost everywhere convergence. Z. Wahrsch. Verw. Gebiete 3 (1964), 75–88.CrossRefGoogle Scholar
3Cwickel, M. and Fefferman, C.. Maximal seminorms on weak L1. Studia Math. 69 (1980), 149–154.CrossRefGoogle Scholar
4Davis, B.. On the integrability of the ergodic maximal function. Studia Math. 73 (1982), 153–167.CrossRefGoogle Scholar
5Dehling, H., Denker, M. and Philipp, W.. Invariance principles for Von Mises and U-statistics. Z. Wahrsch. Verw. Gebiete 67 (1984), 139–167.CrossRefGoogle Scholar
6Gabriel, J. P.. Martingales with a countable filtering index set. Ann. Probab. 5 (1977), 888–898.CrossRefGoogle Scholar
7Gundy, R. F.. On the class L log L, martingales, and singular integrals. Studia Math. 33 (1969), 109–118.CrossRefGoogle Scholar
8Gut, A.. Moments of the maximum of normed partial sums of random variables with multidimensional indices. Z. Wahrsch. Verw. Gebiete 46 (1979), 205–220.CrossRefGoogle Scholar
9Hill, T. P.. Conditional generalizations of strong laws which conclude the partial sums converge almost surely. Ann. Probab. 10 (1982), 828–830.CrossRefGoogle Scholar
10Hoeffding, W.. A class of statistics with asymptotically normal distribution. Ann. Math. Slat. 19 (1948), 293–325.CrossRefGoogle Scholar
11Hunt, R. A.. On L(p, q) spaces. L'enseign. Math. (2) 12 (1966), 249–276.Google Scholar
12Köthe, G.. Topological vector spaces I. (New York: Springer, 1969).Google Scholar
13Marcinkiewicz, J. and Zygmund, A.. Sur les fonctions indépendantes. Fund. Math. 29 (1937), 60–90.CrossRefGoogle Scholar
14McConnell, T. R.. A two-parameter maximal ergodic theorem with dependence. Ann. Probab. (to appear).Google Scholar
15McConnell, T. R.. Triple integration with respect to stable processes (preprint).Google Scholar
16McConnell, T. R.. On the strong maximal function and rearrangements. Studia Math, (to appear).Google Scholar
17McConnell, T. R. and Taqqu, M. S.. Decoupling of Banach-valued multilinear forms in independent, symmetric Banach-valued random variables, Z. Wahrsch. Verw. Gebiete (to appear).Google Scholar
18Rosinski, J. and Woyczynski, W.. On Ito stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals. Ann Probab. 14 (1986), 271–286.CrossRefGoogle Scholar
19Smythe, R. T.. Sums of independent random variables on partially ordered sets. Ann. Probab. 2 (1974), 906–917.CrossRefGoogle Scholar
20Stein, E. M. and Weiss, N.. On the convergence of Poisson integrals. Trans. Amer. Math. Soc, 140 (1969), 35–54.CrossRefGoogle Scholar