  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
$n\to\infty$.
