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(X_i, X_k). Let q_i,  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 Δ_n = ρ(T, T₀) ≤ cβ₃q⁻¹₁ n^−1/7 +cq⁻¹₁∑_{i ≥ 1} q_in^−1/4, where T₀ is the limit statistic and ρ is a Kolmogorov (or uniform) distance.