Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics

Abstract

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 $S$, and let $f_{i_1...i_k}$ be measurable functions from $S^k$ 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 $n\ge 2$, $$P(||\sum _{1\le i_1\ne ...\ne i_k\le n}{f}_{i_1...i_k}(X_{i_1},...,X_{i_k})||\ge t)$$ $$\le C_{k}P(C_k||\sum _{1\le i_1\ne ...\ne i_k\le n}{f}_{i_1...i_k}(X_{i_1}^{(1)},...,X_{i_k}^{(k)})||\ge t).$$ Furthermore, the reverse inequality also holds in the case that the functions $\{f_{i_1...i_k}\}$ satisfy the symmetry condition $$f_{i_1...i_k}(X_{i_1},...,X_{i_k})=f_{i_{\pi (1)}...i_{\pi (k)}}(X_{i_{\pi (1)}},...,X_{i_{\pi (k)}})$$ for all permutations $\pi$ of $\{1,...,k\}$. Note that the expression $i_1\ne ...\ne i_k$ means that $i_r\ne i_s$ for $r\ne s$. Also, $C_k$ is a constant that depends only on $k$.

References in corpus (2)

• Bounds on the tail probability of U-statistics and quadratic forms
• Contraction and decoupling inequalities for multilinear forms and u-statistics