The $\Delta_2$-condition and $\varphi$-families of probability distributions

Abstract

In this paper, we provide some results related to the $\Delta_2$-condition of Musielak-Orlicz functions and $\varphi$-families of probability distributions, which are modeled on Musielak-Orlicz spaces. We show that if two $\varphi$-families are modeled on Musielak-Orlicz spaces generated by Musielak-Orlicz functions satisfying the $\Delta_{2}$-condition, then these $\varphi$-families are equal as sets. We also investigate the behavior of the normalizing function near the boundary of the set on which a $\varphi$-family is defined.