Bounds for Non-Gaussian Approximations of U-Statistics

Let X, X,X 1,...,X n be i.i.d. random variables taking values in a measurable space ( X,B). Consider U-statistics of degree two U=n^-1 (sigma) (phi)(Xi,Xj) with symmetric, degenerate kernel (phi): Xi,Xj. Let U(infinity) = ½ (sigma) qj(tau)-1where {q j } are eigenvalues of the Hilbert–Schmidt operator associated with the kernel (phi)and {(tau) j } are i.i.d. standard normal random variables. If E (phi)2 (x,x)then (delta) := sup | P{U