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.