AbstractWe give counterexamples and a few positive results related to several conjectures of R. Pemantle (Pemantle, J Math Phys 41 (2000), 1371–1390) and D. Wagner (Wagner, Ann Combin 12 (2008), 211–239) concerning negative correlation and log‐concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. (Borcea et al., J Am Math Soc 22 (2009), 521–567). We also give short proofs of a pair of results from (Pemantle, J Math Phys 41 (2000), 1371–1390) and (Borcea et al., J Am Math Soc 22 (2009), 521–567); prove that “almost exchangeable” measures satisfy the “Feder‐Mihail” property, thus providing a “non‐obvious” example of a class of measures for which this important property can be shown to hold; and mention some further questions. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010