On probabilities of large deviations for \(UH\)-statistics
Let \(X_1,...,X_n\) be i.i.d. random variables taking values in a measurable space \((X,{\mathcal X})\). Let \(H\) be a separable Hilbert space. Let \(\Phi=\Phi(x,y):X^2\to H\) be a symmetric function. Denote \( g(x,y)=\Phi(x,y)-\Theta-g_1(x)-g_1(y)\), where \(\Theta={\mathbf E}\Phi(X_1,X_2)\), \(g_1(x)={\mathbf E}(\Phi(X_1,X_2)\mid X_2=x).\) Consider \(UH\)-statistics \( U_n=(n/2)^{-1} \sum_{1\leq i_1<i_2\leq n}\Phi(X_{i_1},X_{i_2})\). The main aim of the paper is to estimate the variable \[ \Delta_{n,2}(H;x)=|{\mathbf P}(|2^{-1}\sqrt{n}(U_n-\Theta)|>x)/ {\mathbf P}(|\tau|>x)-1|, \] where \(\tau\) is a Gaussian random element in \(H\) with the mean value zero and the covariance operator \(S\): \( (Sx,y)=\int_{X}(x,g_1(z))(y,g_1(z)) {\mathbf P}(X_1\in dz)\). Theorem. Assume that \({\mathbf E}|g_1(X_1) |^2 \not=0, {\mathbf E}\exp\{h|g_1(X_1)|^{\alpha}\}<\infty\) for some \(h>0\), \(0<\alpha\leq 1/2\), and there exist constants \(\gamma\geq 2\) and \(C>0\) such that \({\mathbf E}|g(X_1,X_2) |^k \leq C^k k^{\gamma k}, k=1,2,...\). Then there exist positive constants \(\delta\) and \(\varepsilon\), independent of \(n\) and \(x\) such that the estimate \[ \Delta_{n,2}(H;x)\leq \varepsilon (1+x^3+\log n)/\sqrt{n} \] holds in the region \(0\leq x \leq \delta n^{1/(4\gamma-2)}\).
scientific article; zbMATH DE number 939516