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## Asymptotic rate of convergence in the degenerate U-statistics of second order

### Volume 90 / 2010

Banach Center Publications 90 (2010), 275-284 MSC: Primary 60F99; Secondary 46A32. DOI: 10.4064/bc90-0-17

#### 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 $\beta':=\mathbf{E}\,|h(X,X_1)|^{3}+\mathbf{E}\,|h(X,X_1)|^{18/5}+ \mathbf{E}\,|g(X)|^{3}+ \mathbf{E}\,|g(X)|^{18/5}+1< \infty$.

#### Authors

• Olga YanushkevichieneVilnius Pedagogical University
Studentu 39
Vilnius LT-08106, Lithuania
and
Institute of Mathematics and Informatics