**Stephen Montgomery-Smith and Victor de la Peña, Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics.**
*Annals Prob. ***23**, (1995), 806-816.
In this paper the following result, which allows one to decouple U-Statistics in tail probability, is proved in full generality.
Theorem 1. Let \(X_i\) be a sequence of independent random variables taking values in a measure space \(\mathcal S\), and let \(f_{i_1,\dots,i_k}\) be measurable functions from \(\mathcal S^k\) to a Banach space \(B\). Let \((X_i^{(j)})\) be independent copies of \((X_i)\). The following inequality holds for all \(t \ge 0\) and all \(n \ge 2\)
\[ P\left(\left\|\sum_{1 \le i_1 \ne \cdots \ne i_k \le n} f_{i_1,\dots,i_k}(X_{i_1},\dots,X_{i_k})\right\| \ge t\right)
\le C_k
P\left(\left\|C_k \sum_{1 \le i_1 \ne \cdots \ne i_k \le n} f_{i_1,\dots,i_k}(X_{i_1}^{(1)},\dots,X_{i_k}^{(k)})\right\| \ge t\right) .\]
Furthermore, the reverse inequality also holds in the case that the functions \(\{f_{i_1,\dots,i_k}\}\) satisfy the symmetry condition
\[ f_{i_1,\dots,i_k}(X_{i_1},\dots,X_{i_k}) = f_{i_{\pi(1)},\dots,i_{\pi(k)}}(X_{i_{\pi(1)}},\dots,X_{i_{\pi(k)}}) \]
for all permutations \(\pi\) of \(\{1,\dots,k\}\). Note that the expression \(i_1 \ne \cdots \ne i_k\) means that \(i_r \ne i_s\) for \(r \ne s\). Also, \(C_k\) is a constant that depends only on \(k\).
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