= Positive association of random variables
{title2=$\operatorname{Cov}(f(X),g(X))\geq0$}
A random vector $X$ is positively associated if $\operatorname{Cov}(f(X),g(X))\geq0$ 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 $\mathbb P(\bigcap_iE_i)\geq\prod_i\mathbb P(E_i)$ 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 https://www.stat.cmu.edu/~brian/720/week02/esary-proschan-walkup-1967.pdf[Esary, Proschan and Walkup (1967)].
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