Let K be a probability density function on Rn, and f and g be two increasing functions on Rn. The famous FKG inequality asserts that if K satisfies an MTP2 condition then the corresponding correlation between f and g is nonnegative. From another perspective, the FKG inequality gives us a view of some of the probabilistic behavior of any probability density function which satisfies the MTP2 hypothesis. In this talk, I will show that much more probabilistic information beyond the FKG inequality is satisfied by the function K, and this too without any additional (MTP-type or other) assumptions. Specifically, we shall derive from the MTP2 condition a hierarchy of FKG-type inequalities of which the classical FKG inequality appears to be a simple instance.
This talk is based on joint research with Helene Massam.