Optimal bounds in non-Gaussian limit theorems for U-statistics

Let X, X-1, X-2,... be i.i.d. random variables taking values in a measurable space H. Let phi(x, y) and phi(1)(x) denote measurable functions of the arguments x, y is an element of H. Assuming that the kernel phi is symmetric and that E phi(x, X) = 0, for all x, and E phi(1)(X) = 0, we consider U-statistics of type [GRAPHICS] It is known that the conditions E phi(2)(X, X-1) < infinity and E phi(1)(2)(X) < infinity imply that the distribution function of T, say F, has a limit, say F-0, which can be described in terms of the eigenvalues of the Hilbert-Schmidt operator associated with the kernel phi(x, y). Under optimal moment conditions, we prove that [GRAPHICS] provided that at least nine eigenvalues of the operator do not vanish. Here F-1 denotes an Edgeworth-type correction. We provide explicit bounds for Delta(N) and for the concentration functions of statistics of type T.