The ${\Delta }_2$-condition and $\phi$-families of probability distributions

Abstract

In this paper, we provide some results related to the ${\Delta }_2$-condition of Musielak-Orlicz functions and $\phi$-families of probability distributions, which are modeled on Musielak-Orlicz spaces. We show that if two $\phi$-families are modeled on Musielak-Orlicz spaces generated by Musielak-Orlicz functions satisfying the ${\Delta }_2$-condition, then these $\phi$-families are equal as sets. We also investigate the behavior of the normalizing function near the boundary of the set on which a $\phi$-family is defined.