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On a normal approximation of U-statistics. (English. Russian original) Zbl 0993.60021
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 $$\sup_{x>0} (x^{5/3} P(|h(X_1, X_2)|>x)).$$

##### MSC:
 60F05 Central limit and other weak theorems
##### Keywords:
U-statistic; Berry-Esseen inequality; normal approximation
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