The Δ2-condition and φ-families of probability distributions

Abstract

In this paper, we provide some results related to the Δ2- condition of Musielak-Orlicz functions and φ-families of probability distributions, which are modeled on Musielak-Orlicz spaces. We show that if two φ-families are modeled on Musielak-Orlicz spaces generated by Musielak-Orlicz functions satisfying the Δ2-condition, then these φ-families are equal as sets. We also investigate the behavior of the normalizing function near the boundary of the set on which a φ-family is defined. © 2013 Springer-Verlag.