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Let be a U-statistic based on a kernel function h(x 1, x 2) and independent and identically distributed samples {X n ; n ≥ 1}. The author obtained a weak invariance principle and a strong functional limit theorem for the products of when may be infinite. Moreover, the author also applies the results to the sample variance and the Gini's mean difference, and obtain the limiting properties of their products. [ABSTRACT FROM AUTHOR]

Weak invariance principles are proved for regression rank statistics. As a consequence limit theorems for max- and Lp-functionals of partial sums of vectors of simple linear rank statistics are obtained. The results are useful in change point analysis, particularly in justification of application of permutation arguments, see Antoch and Hušková [Antoch, J.; Hušková, M. Detection of Structural Changes in Regression. Tatra Mountains Publications, 2003, 26, 1–15] and Hušková and Picek [Hušková, M.; Picek, J. M-tests for detection of structural changes in regression. In Statistical Data Analysis Based on the L1-Norm and Related Methods; Dodge, Y., Ed.; Birhäuser: Basel, 2002; 213–229]. The results of Hušková [Hušková, M. Limit theorems for rank statistics. Statist. Probab. Letters 1997, 32, 45–55] are generalized. [ABSTRACT FROM AUTHOR]