Negative correlation and log-concavity

OF THE DISSERTATION Negative correlation and log-concavity by Michael Neiman Dissertation Director: Jeff Kahn This thesis is concerned with negative correlation and log-concavity properties and relations between them, with much of our motivation provided by [40], [46], and [12]. Our main results include a proof that “almost exchangeable” measures satisfy the “FederMihail” property; counterexamples and a few positive results related to several conjectures of Pemantle [40], Wagner [46], and Choe and Wagner [7] concerning negative correlation and log-concavity properties for probability measures and relations between them; a proof that a conditional version of the “antipodal pairs property” implies a strong form of log-concavity, which yields some partial results on a well-known conjecture of Mason [38]; a proof that “competing urn” measures satisfy “conditional negative association”; and proofs that certain classes of measures introduced by Srinivasan [42] and Pemantle [40] satisfy a strong form of negative association.