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 $$k$$. (tex, dvi, ps, pdf.)

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