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Abstract

Let $X,X_1,\ldots ,X_n$ be independent identically distributed random variables taking values in a measurable space $(\Theta, \Re)$. Let $h(x,y)$ and $g(x)$ be real valued measurable functions of the arguments $x, y\in\Theta$ and let $h(x,y)$ be symmetric. We consider U-statistics of the type $$ T(X_1,\dots ,X_n)=n^{-1} \sum_{1\leq i< k\leq n}h(X_i,X_k) + n^{-1/2} \sum_{1\leq i\leq n}g(X_i). $$ Let $q_i \ (i\geq1)$ be eigenvalues of the Hilbert-Schmidt operator associated with the kernel $h(x,y)$, and $q_1$ be the largest in absolute value one. We prove that $$ \Delta_{n}=\rho(T(X_1,\dots ,X_n),T(G_1,\dots ,G_n))\leq \frac{c\beta^{\prime 1/6}}{\sqrt{|q_1|}n^{1/12}}, $$ where $G_i, \ 1\leq i\leq n$, are i.i.d. Gaussian random vectors, $\rho$ is the Kolmogorov (or uniform) distance and $\beta':=\mathbf{E}\,|h(X,X_1)|^{3}+\mathbf{E}\,|h(X,X_1)|^{18/5}+ \mathbf{E}\,|g(X)|^{3}+ \mathbf{E}\,|g(X)|^{18/5}+1< \infty$.