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Normal approximation of $$U$$-statistics in Hilbert space. (English. Russian original) Zbl 0889.60004
Theory Probab. Appl. 41, No. 3, 405-424 (1996); translation from Teor. Veroyatn. Primen. 41, No. 3, 481-504 (1996).
Let $$X_1,\dots,X_n$$ be independent identically distributed random variables with values in a measurable space $$(M,{\mathcal M})$$. Let $$\Phi$$ be a symmetric function from $$M\times M$$ into a real separable Hilbert space $$(H,|\cdot|)$$ such that $$E\Phi (X_1,X_2)=0$$ and $$E|\Phi (X_1,X_2)|^2<\infty$$. Also, let $$g_1(x):=E[\Phi (X_1,X_2)\mid X_1=x]$$ and $$g_2(x,y):=\Phi (x,y)- g_1(x)-g_2(y)$$. Suppose that $$\sigma^2:=E|g_1(X_1)|^2$$ is finite and non-zero. Then the covariance operator $$V$$ of $$g_1(X_1)$$ is defined and $$\sigma^2= \text{ tr}V$$. Denote by $$\sigma_1^2\geq \sigma_2^2\geq\cdots$$ the eigenvalues of $$V$$. Let $$U_n:={n\choose 2}^{-1}\sum_{1\leq i<j\leq n}\Phi (X_i,X_j)$$ be a $$U$$-statistic, and let $$Y$$ be a Gaussian $$H$$-valued random variable with mean zero and covariance operator $$\sigma^{-2}V$$. By the central limit theorem for $$U$$-statistics, if $$E|g_1(X_1)|^2 <\infty$$ and $$E|g_2(X_1,X_2)|^{4/3}<\infty$$, then $\Delta_n(a):= \sup_r|\text{Pr}(\{|2^{-1}n^{1/2}\sigma^{-1}U_n-a|<r\})-\text{Pr} (\{|Y-a|<r\})|\to 0$ as $$n\to\infty$$. It is proved that there exists an absolute constant $$c$$ such that, for all $$n\geq 2$$ and $$a\in H$$, $\Delta_n(a)\leq c(1+|a|^3)\sigma^4(\sigma_1^{-1}\cdots\sigma_8^{-1}) \sigma_9^{-2}\beta^2n^{-1/2}$ provided $$\beta:=E|\Phi (X_1,X_2)|^3<\infty$$.

##### MSC:
 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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