Normal Approximation of U-Statistics in Hilbert Space

Abstract

Let {Un}, n=1,2,..., be Hilbert space H-valued U-statistics with kernel Φ(⋅,⋅), corresponding to a sequence of observations (random variables) $X_1,X_2,\ldots\ $. The rate of convergence on balls in the central limit theorem for {Un} is investigated. The obtained estimate is of order n−1/2 and depends explicitly on E∥Φ(X1,X2)∥3 and on the trace and the first nine eigenvalues of the covariance operator of E(Φ(X1,X2)|X1).