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April, 1995 Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics
Victor H. de la Pena, S. J. Montgomery-Smith
Ann. Probab. 23(2): 806-816 (April, 1995). DOI: 10.1214/aop/1176988291

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 $(1,\ldots,k)$.

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Victor H. de la Pena. S. J. Montgomery-Smith. "Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics." Ann. Probab. 23 (2) 806 - 816, April, 1995. https://doi.org/10.1214/aop/1176988291

Information

Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0827.60014
MathSciNet: MR1334173
Digital Object Identifier: 10.1214/aop/1176988291

Subjects:
Primary: 60E15
Secondary: 60D05

Keywords: $U$-statistics , Decoupling

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 2 • April, 1995
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