U-statistics on associated random variables

• Published in: Journal of statistical planning and inference Vol. 119; no. 1; pp. 1 - 15
• Main Author: Christofides, Tasos C.
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 15-01-2004; New York,NY Elsevier Science; Amsterdam
• Subjects: Associated random variables; Demimartingales; Distribution theory; Exact sciences and technology; Limit theorems; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic processes; U-statistics; 60E15; Associated random variables; Demimartingales; 60G42; U-statistics; 60F15; Demimartingale; U statistic; Associated random variable; Independent variable; Probability theory; Law of large numbers
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/S0378-3758(02)00418-4

We consider the class of U-statistics which are based on a collection of associated random variables. Since independent random variables are associated, the class of U-statistics introduced by Hoeffding (Ann. Math. Statist. 19, 293–325) is included as a special case. Since many of the theoretical results which are valid for Hoeffding's U-statistics depend on the assumption of independence, the theory known cannot automatically be applied to the case of U-statistics based on associated random variables. In this paper, we prove a strong law of large numbers for this class of statistics in the case where the “kernel” of the U-statistic belongs to a large family of functions.