## The Annals of Probability

### Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics

#### Abstract

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 $\{X_j\}$ be a sequence of independent random variables on a measurable space $(\mathscr{J}, S)$ and let $\{X^{(j)}_i\}, j = 1,\ldots,k$, be $k$ independent copies of $\{X_i\}$. Let $f_{i_1i_2\ldots i_k}$ be families of functions of $k$ variables taking $(S \times \cdots \times S)$ into a Banach space $(B, \|\cdots\|)$. Then, for all $n \geq k \geq 2, t > 0$, there exist numerical constants $C_k$ depending on $k$ only so that $P\bigg(\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(1)}_{i_2}, \ldots, X^{(1)}_{i_k})\big\|\geq t\bigg)$ $\leq C_kP\bigg(C_k\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(2)}_{i_2}, \ldots, X^{(k)}_{i_k})\big\|\geq t\bigg).$ The reverse bound holds if, in addition, the following symmetry condition holds almost surely: $f_{i_1i_2\cdots i_k}(X_{i_1}, X_{i_2},\ldots,X_{i_k}) = f_{i_{\pi(1)}i_{\pi(2)}\cdots i_{\pi(k)}} (X_{i_{\pi(1)}, X_{i_{\pi(2)}}, \ldots,X_{i_{\pi(k)}}),$ for all permutations $\pi$ of $(1,\ldots,k)$.

#### Article information

Source
Ann. Probab., Volume 23, Number 2 (1995), 806-816.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988291

Digital Object Identifier
doi:10.1214/aop/1176988291

Mathematical Reviews number (MathSciNet)
MR1334173

Zentralblatt MATH identifier
0827.60014

JSTOR
$U$-statistics decoupling
de la Pena, Victor H.; Montgomery-Smith, S. J. Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics. Ann. Probab. 23 (1995), no. 2, 806--816. doi:10.1214/aop/1176988291. https://projecteuclid.org/euclid.aop/1176988291