Maximal inequalities and convergence results for generalized U- statistics
type: Item
instance of: scholarly article
title: Maximal inequalities and convergence results for generalized U- statistics
zbMATH Open document ID: 0693.62022
DOI: 10.1016/0378-3758(90)90048-Y
author: Tasos C. Christofides
author: Robert J. Serfling
published in: Journal of Statistical Planning and Inference
publication date: 1990-01-01T00:00:00Z
review text: The authors present some key probability inequalities and convergence results for multisample, or `generalized U-statistics'. They exploit the reverse martingale structure of generalized U-statistics to obtain a useful Kolmogorov-type inequality. In addition they obtain a bound for the rate of convergence in the strong law of large numbers for generalized U-statistics, extending the one-sample result of \textit{W. F. Grams} and \textit{R. J. Serfling} [Ann. of Statist. 1, 153-160 (1973; Zbl 0322.62053)]. Like \textit{P. K. Sen} [Ann. Probab. 5, 287-290 (1977; Zbl 0362.60019 )] they exhibit another class of generalized U-statistics for which the strong law holds under merely a first moment condition. Finally using \textit{W. Hoeffding}'s [The strong law of large numbers for U-statistics. Univ. North Carolina Inst. Stat. Mimeo Ser. No.302 (1961)] forward martingale representation they establish an invariance principle for generalized U-statistics.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 62G99
zbMATH DE Number: 4135207
zbMATH Keywords: generalized U-statistics
zbMATH Keywords: reverse martingale structure
zbMATH Keywords: Kolmogorov-type inequality
zbMATH Keywords: rate of convergence
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: first moment condition
zbMATH Keywords: forward martingale representation
zbMATH Keywords: invariance principle
reviewed by: Adhir K. Basu
MaRDI profile type: Publication
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full work available at URL: 0378-3758(90)90048-y
OpenAlex ID: W2094065723
description: scientific article; zbMATH DE number 4135207