On the central limit theorem for \(U\)-statistics under absolute regularity
The author proves a central limit theorem for \(U\)-statistics under absolute regularity \((\beta\)-mixing). This improves the conditions of \textit{M. Denker} and \textit{G. Keller} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 64, 505-522 (1983; Zbl 0519.60028)] and \textit{K.-i. Yoshihara} [ibid. 35, 237-252 (1976; Zbl 0314.60028)]. Proving this CLT the author uses recent improved limit theorems of \textit{E. Rio} [Ann. Inst. Henri Poincaré, Probab. Stat. 29, No. 4, 587-597 (1993; Zbl 0798.60027)] and \textit{P. Doukhan}, \textit{P. Massart} and \textit{E. Rio} [ibid. 30, No. 1, 63- 82 (1994; Zbl 0790.60037)] for a strictly stationary process and some bound on the remainder terms.
scientific article; zbMATH DE number 820291