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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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