Abstract
A probability measure $\mu$ on the lattice $2^{[n]}$ is said to be positively associated if any two increasing functions on the lattice are positively correlated with respect to $\mu$. Pemantle asked whether, in order to establish positive association for a given $\mu$, it might be sufficient to show positive correlation only for pairs of functions which depend on disjoint subsets of the ground set $[n]$. We answer Pemantle's question in the negative, by exhibiting a measure which gives positive correlation for pairs satisfying Pemantle's condition but not for general pairs of increasing functions.
Citation
Nicholas Weininger. "Positive correlation for increasing events with disjoint dependencies does not imply positive correlation for all increasing events." Electron. Commun. Probab. 8 99 - 101, 2003. https://doi.org/10.1214/ECP.v8-1078
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