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Asymptotics of U-statistics and von Mises functionals. (English. Russian original) Zbl 0533.60043
Sov. Math., Dokl. 27, 303-308 (1983); translation from Dokl. Akad. Nauk SSSR 269, 265-269 (1983).
The paper contains announcements of results which sharpen the moment conditions in results of W. R. van Zwet and R. Helmers [Math. Cent., Amst., Afd. Math. Stat. SW 75/81 (1981; Zbl 0476.62025)] and the reviewer [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 333-355 (1979; Zbl 0405.60009) and Ann. Probab. 9, 852-859 (1981; Zbl 0473.60009)]. In the result on the Berry-Esseen theorem for U-statistics of degree two and higher the author claims that a moment of order 5/3 for the kernel suffices. Further refinements concerning the rate of convergence for degenerated U-statistics are a Berry-Esseen bound with a moment of order 3/2 for the r.v. \(\Phi(X_ 1,X_ 1)\) and a rate \(n^{-1}\ln n\) when \(E| \Phi(X_ 1,X_ 2)|^ 4+E| \Phi(X_ 1,X_ 1)|^ 2<\infty\).
The refinement claimed for the Berry-Esseen bound in Banach spaces of type 2 of the reviewer with \(E\| X_ 1\|^ 3<\infty\) and a three times differentiable functional is rather strong. At present the reviewer does not know of any proof being published of these results.
Reviewer: F.Götze

60F17 Functional limit theorems; invariance principles
62E20 Asymptotic distribution theory in statistics
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60E10 Characteristic functions; other transforms
60F05 Central limit and other weak theorems