On a normal approximation of U-statistics. (English. Russian original) Zbl 0993.60021

Borovskikh, Yu. V.

Theory Probab. Appl. 45, No. 3, 406-423 (2000); translation from Teor. Veroyatn. Primen. 45, No. 3, 469-488 (2000).

Let \(U_n\) be a non-degenerate \(U\)-statistic of degree 2 based on independent and identically distributed random variables \(X_1, \dots , X_n.\) Let \(\Phi\) be the kernel and assume \(\Phi\) is symmetric. Let \(\theta = E(\Phi(X_1,X_2))\). Define the functions \(g(x) = E \Phi(X_1,x)\) and \(h(x,y) = \Phi(x,y) - \theta-g(x)-g(y).\) This paper investigates the rate of convergence in the central limit theorem for \(U_n.\) The result obtained implies that a rate of order \(n^{-1/2}\) depends only on the third moment \(E|g(X_1)|^3\) and the weak moment \(\sup_{x>0} (x^{5/3} P(|h(X_1, X_2)|>x)).\)

Reviewer: Neville Weber (Sydney)

MSC:

60F05

Central limit and other weak theorems

Keywords:

U-statistic; Berry-Esseen inequality; normal approximation