On the rate of convergence of degenerate U-statistics

Abstract

Let X, X1, X2, ... be independent identically distributed random variables taking values in a measurable space (Ω, ℜ). Let h(x, y) be real valued measurable symmetric function of the arguments x, y ∈ ℜ. Assume that Eh(x, X)= 0, for all x. We consider U-statistics of type T = n−1 ∑1 ≤ i< k ≤ n h(Xi, Xk). Let qi, i ≥ 1 be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x, y) and q1 be the largest eigenvalue. Under the condition β3 := E|h(X, X1)|3 <∞, we prove that Δn = ρ(T, T0) ≤ cβ3q−11 n−1/7 +cq−11∑i ≥ 1 qin−1/4, where T0 is the limit statistic and ρ is a Kolmogorov (or uniform) distance.

Keywords

• U-statistics of second degree
• rate of convergence
• degenerate U-statistics