The rate of convergence in the central limit theorem for \(UH\)-statistics. (English. Russian original) Zbl 0738.60005

Korolyuk, V. S.; Borovskikh, Yu. V.

Theory Probab. Math. Stat. 43, 81-86 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 72-78 (1990).

Summary: The rate of convergence is estimated for the distributions of nondegenerate \(U\)-statistics with values in a separable real Hilbert space. For such \(UH\)-statistics of unit rank the problem is reduced to the use of asymptotic methods of analyzing the probability of hitting a ball with nonzero random center in the Hilbert space. An estimate of the rate of convergence of order \(o(n^{-1/2})\) is given.

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05	Central limit and other weak theorems
62F10	Point estimation

Keywords:

rate of convergence; \(UH\)-statistics; asymptotic methods; probability of hitting a ball