Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics
Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics
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多元 $U$ 统计尾部概率的解耦不等式
DOI：
10.1214/aop/1176988291
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发表时间：
1993
影响因子：
2.3
通讯作者：
S. Montgomery
中科院分区：
文献类型：
--
作者：
V. Peña;S. Montgomery
在本文中，我们提出了一个解耦不等式，表明多元 $U$ 统计量可以作为（条件）独立随机变量的总和进行研究。这一结果对概率和统计的多个领域具有重要意义，包括随机图和多重随机积分的研究。更准确地说，我们得到以下结果：令 $\{X_j\}$ 为可测空间 $(\mathscr{J}, S)$ 上的独立随机变量序列，并令 $\{X^{(j)}_i\}, j = 1,\ldots,k$ 为 $\{X_i\}$ 的 $k$ 个独立副本。设 $f_{i_1i_2\ldots i_k}$ 为 $k$ 个变量的函数族，将 $(S \times \cdots \time S)$ 放入 Banach 空间 $(B, \|\cdots\|)$ 中。然后，对于所有 $n \geq k \geq 2, t > 0$，存在仅取决于 $k$ 的数值常量 $C_k$，因此 $P\bigg(\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(1)}_{i_2}, \ldots, X^{(1)}_{i_k})\big\|\geq t\bigg)$ $\leq C_kP\bigg(C_k\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(2)}_{i_2}, \ldots, X^{(k)}_{i_k})\big\|\geq t\bigg).$ 如果此外，以下对称条件几乎肯定成立，则反向界限成立： $f_{i_1i_2\cdots i_k}(X_{i_1}, X_{i_2},\ldots,X_{i_k}) = f_{i_{\pi(1)}i_{\pi(2)}\cdots i_{\pi(k)}} (X_{i_{\pi(1)}, X_{i_{\pi(2)}}, \ldots,X_{i_{\pi(k)}}),$ 对于所有排列 $\pi$ $(1,\ldots,k)$。
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 $(1,\ldots,k)$.