Geodesic
The central limit theorem for nonhomogeneous $U$-statistics of finitely dependent random variables
Sbornik. Mathematics, Tome 27 (1975) no. 4, pp. 554-563 
V. G. Mikhailov. The central limit theorem for nonhomogeneous $U$-statistics of finitely dependent random variables. Sbornik. Mathematics, Tome 27 (1975) no. 4, pp. 554-563. http://geodesic.mathdoc.fr/item/SM_1975_27_4_a6/
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     title = {The central limit theorem for nonhomogeneous $U$-statistics of finitely dependent random variables},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper the notion of homogeneous vector-valued $U$-statistic is introduced and a central limit theorem is proved for such statistics of a sequence of finitely dependent random variables, which generalizes analogous results for ordinary $U$-statistics. The convergence of finite-dimensional distributions of some random processes generated by $U$-statistics is also studied. Bibliography: 5 titles.

[1] W. Hoeffding, “A class of statistics with asymptotically normal distribution”, Ann. Math. Statist., 19:3 (1948), 293–325 | DOI | MR | Zbl

[2] R. G. Miller, P. K. Sen, “Weak convergence of $U$-statistics and Von-Mises 'differentiable statistical functions”, Ann. Math. Statist., 43:1 (1972), 31–43 | DOI | MR | Zbl

[3] P. K. Sen, “Weak convergence of generalized $U$-statistics”, Ann. Prob., 2:1 (1974), 90–102 | DOI | MR | Zbl

[4] B. Abdalimov, “Asimptoticheskaya normalnost $U$-statistik dlya $s$-zavisimykh velichin”, Sluchainye protsessy i smezhnye voprosy, ch. 1, izd-vo «Fan», Tashkent, 1970, 26–30 | MR

[5] B. Abdalimov, T. L. Malevich, “O tsentralnoi predelnoi teoreme dlya $U$-statistik”, Sluchainye protsessy i statisticheskie vyvody, vyp. II, izd-vo «Fan», Tashkent, 1972, 3–9