On Berry-Esséen rates for \(m\)-dependent \(U\)-statistics
The author considers the \(U\)-statistics with strictly stationary \(m\)-dependent random variables \(X_1,X_2,\dots\) with \[ \sigma^2= Eg^2(X)+2\bigl[ Eg(X_1)g(X_2)+ \cdots+ Eg(X_1)g (X_{m+1})\bigr]>0, \] where \[ U_n={n\choose 2}^{-1} \sum_{1\leq i<j\leq n}h(X_i,X_j),\quad \theta=Eh(X_1,X_{m+1}) \quad\text{and}\quad g(y)= E\bigl[h(X,y)- \theta\bigr]. \] Here the kernel of the \(U\)-statistics \(h(x,y)\) is a real-valued symmetric function of two variables. The main theorem gives the best Berry-Esséen bound as follows, for \(n\geq 3\), \[ \sum_x\left| P \left({\sqrt n(U_n-EU_n)\over 2\sigma}\leq x\right)-\Phi(x) \right|\leq Cn^{-1/2} \bigl(\rho+ \lambda_{5/3}+ (\rho\lambda_{3/2})^{2/3} \bigr), \] where \(\lambda_\nu= \sigma^{-\nu} \max_{2\leq j\leq m+1} E| h(X_1,X_j) |^\nu\). This result is similar to that for the \(U\)-statistics based on independent random variables.
scientific article; zbMATH DE number 1228679
scientific article