Decoupling and Khintchine's Inequalities for U-Statistics

In this paper we introduce a fairly general decoupling inequality for U-statistics. Let ᵢ\ be a sequence of independent random variables in a measurable space (S, J), and let ̃ᵢ\ be an independent copy of ᵢ\. Let Φ(x) be any convex increasing function for x ≥ 0. Let Πᵢⱼ be families of functions of two variables taking (S × S) into a Banach space (D, \|·\|). If the fᵢⱼ ∈ Πᵢⱼ are Bochner integrable and max1≤ i≠ j≤ n EΦ(_fᵢⱼ∈Πᵢⱼ\|fᵢⱼ(Xᵢ, Xⱼ)\|) < ∞, then, under measurability conditions, $EΦ(f∈Π\|∑1≤ i≠ j≤ n fᵢⱼ(X_i, X_j)\|) ≤ EΦ(8 f∈Π\|∑1≤ i≠ j≤ n fᵢⱼ(X_i, X̃_j)\|),$ where $f = (fᵢⱼ, 1 ≤ i ≠ j ≤ n)$ and $Π = (Πᵢⱼ, 1 ≤ i ≠ j ≤ n)$. In the case where $Π$ is a family of functions of two variables satisfying $fᵢⱼ = fⱼᵢ$ and $fᵢⱼ(X_i, X_j) = fᵢⱼ(X_j, X_i)$, the reverse inequality holds (with a different constant). As a corollary, we extend Khintchine's inequality for quadratic forms to the case of degenerate $U$-statistics. A new maximal inequality for degenerate $U$-statistics is also obtained. The multivariate extension is provided.