 Positive association of random variables

A random vector $X$ is positively associated if $\mathrm{Cov}(f(X),g(X))\ge 0$ for every pair of coordinatewise nondecreasing bounded functions $f,g$. The same inequality holds when both are nonincreasing, by negating them. Repeatedly applying it to products of nonnegative increasing indicators gives $\mathbb{P}({\bigcap }_i{E}_i)\ge {\prod }_i\mathbb{P}(E_i)$ for increasing events; the same is true for decreasing events. Thus associated rejection indicators have familywise error rate no greater than independent indicators with the same marginals. Pairwise positive correlation coefficients alone do not give this stronger property. The classical definition is in Esary, Proschan and Walkup (1967).