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

Abstract

Let X, X1, X2, . . . be i.i.d. random variables taking values in a measurable space script X. Let φ(x,y) and φ1(x) denote measurable functions of the arguments x, y ∈ script X. Assuming that the kernel φ is symmetric and that Eφ(x, X) = 0, for all x, and Eφ1(X) = 0, we consider U-statistics of type T = N-1 Σ φ(Xj, Xk) + N-1/2 Σ φ1(Xj). 1≤j < ∞ and Eφ21(X) < ∞ imply that the distribution function of T, say F, has a limit, say F0, which can be described in terms of the eigenvalues of the Hilbert-Schmidt operator associated with the kernel φ(x, y). Under optimal moment conditions, we prove that ΔN = sup|F(x) - F0(X) - F1(x)| = script O(N-1), x provided that at least nine eigenvalues of the operator do not vanish. Here F1 denotes an Edgeworth-type correction. We provide explicit bounds for ΔN and for the concentration functions of statistics of type T.