Decoupling Inequalities for the Tail Probabilities of Multivariate U-Statistics

In this paper we present a decoupling inequality that shows that multivariate U-statistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration. More precisely, we get the following result: Let ⱼ\ be a sequence of independent random variables on a measurable space (J, S) and let ⁽ʲ⁾ᵢ\, j = 1,…,k, be k independent copies of ᵢ\. Let fi₁i₂… iₖ be families of functions of k variables taking (S × ⋯ × S) into a Banach space (B, \|⋯\|). Then, for all n ≥ k ≥ 2, t > 0, there exist numerical constants Cₖ depending on k only so that P(\|∑1≤ i₁≠ i₂≠⋯≠ iₖ≤ n fi₁⋯ iₖ(X⁽¹⁾i₁, X⁽¹⁾i₂, …, X⁽¹⁾iₖ)\|≥ t) ≤ CₖP(Cₖ\|∑1≤ i₁≠ i₂≠⋯≠ iₖ≤ n fi₁⋯ iₖ(X⁽¹⁾i₁, X⁽²⁾i₂, …, X⁽ᵏ⁾iₖ)\|≥ t). The reverse bound holds if, in addition, the following symmetry condition holds almost surely: fi₁i₂⋯ iₖ(Xi₁, Xi₂,…,Xiₖ) = f_iπ(1)iπ(2)⋯ iπ(k) (X_iπ(1), X_iπ(2), …,X_iπ(k)), for all permutations π of (1,…,k).