论文标题
超平方中随机点的相关性不等式，有一些影响
A correlation inequality for random points in a hypercube with some implications
论文作者
论文摘要
令$ \ prec $为$ \ mathbb {r}^k $上的产品订单，并假设$ x_1，x_2，\ ldots，x_n $（$ n \ geq3 $）是i.i.d.随机向量均匀分布在单位HyperCube $ [0,1]^K $中。令$ s $为$ \ mathbb {r}^k $中的（随机）向量集，$ \ {x_3，..，x_n \} $中的所有向量，让$ w $是$ \ in $ \ \ \ \ \ \ \ \ \ \ x_3，x___________________________________________________________________________________________________________________________________________________________________________________________。这项工作的主要结果是相关性不等式\ begin {equation*} p（x_2 \ in w | x_1 \ in W）\ leq p（x_2 \ in w | x_1 \ in s）\，.。 \ end {equation*}对于每$ 1 \ leq i \ leq i \ leq n $ let $ e_ {i，n} $是$ x_i $不是$ \ \ \ {x_1，\ ldots，x_n \} $的其他矢量的$ x_i $不是$ x_i $。主要不平等得出的结果证明了事件$ e_ {1，n} $和$ e_ {2，n} $在$ n \ to \ histty $上是渐近独立的。此外，我们得出了一个相关的组合公式，用于总和$ \ sum_ {i = 1}^n \ textbf {1} _ {e_ {e_ {i，n}} $，即。 $ \ {\ textbf {1} _ {e_ {e_ {i，n}}; 1 \ leq i \ leq n \} $在$ n \ to \ infty $上是渐近正常的。
Let $\prec$ be the product order on $\mathbb{R}^k$ and assume that $X_1,X_2,\ldots,X_n$ ($n\geq3$) are i.i.d. random vectors distributed uniformly in the unit hypercube $[0,1]^k$. Let $S$ be the (random) set of vectors in $\mathbb{R}^k$ that $\prec$-dominate all vectors in $\{X_3,..,X_n\}$, and let $W$ be the set of vectors that are not $\prec$-dominated by any vector in $\{X_3,..,X_n\}$. The main result of this work is the correlation inequality \begin{equation*} P(X_2\in W|X_1\in W)\leq P(X_2\in W|X_1\in S)\,. \end{equation*} For every $1\leq i \leq n$ let $E_{i,n}$ be the event that $X_i$ is not $\prec$-dominated by any of the other vectors in $\{X_1,\ldots,X_n\}$. The main inequality yields an elementary proof for the result that the events $E_{1,n}$ and $E_{2,n}$ are asymptotically independent as $n\to\infty$. Furthermore, we derive a related combinatorial formula for the variance of the sum $\sum_{i=1}^n \textbf{1}_{E_{i,n}}$, i.e. the number of maxima under the product order $\prec$, and show that certain linear functionals of partial sums of $\{\textbf{1}_{E_{i,n}};1\leq i\leq n\}$ are asymptotically normal as $n\to\infty$.
