Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics by

In this paper we present a decoupling inequality that shows that multivariate Ustatistics 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 random graphs and multiple stochastic integration. More precisely, we get the following result: Theorem 1. Let {Xj} be a sequence of independent random variables in a measurable space (S, S), and let {X i }, j = 1, ..., k be k independent copies of {Xi}. 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 6=i2 6=... 6=ik≤n fi1...ik(X (1) i1 , X (1) i2 , ..., X (1) ik )|| ≥ t) ≤ CkP (Ck|| ∑ 1≤i1 6=i2 6=... 6=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) = fiπ(1)iπ(2)...iπ(k)(Xiπ(1) , Xiπ(2) , ..., Xiπ(k)), for all permutations π of (1, ..., k).