The functional limit theorem for the canonical \(U\)-processes defined on dependent trials
This paper provides a functional limit theorem for canonical \(U\)-statistics of arbitrary order based on a stationary sequence of \(\varphi\)-mixing random variables. The result complements the weak limit theorem for canonical \(U\)-statistics of \textit{I. S. Borisov} and \textit{N. V. Volod'ko} [Mat. Tr. 11, No. 1, 25--48 (2008); translation in Sib. Adv. Math. 18, No. 4, 242--257 (2008; Zbl 1249.60027)]. The proof relies on an orthonormal representation of the kernel function associated with the statistic and the limit distribution is described in terms of a linear combination of products of Hermite polynomials evaluated at a sequence of dependent Wiener processes with a known covariance structure.
scientific article; zbMATH DE number 5963627