A correlation inequality for random points in a hypercube with some implications

Abstract

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 $n\to\infty$.