The $Δ_2$-condition and $φ$-families of probability distributions

In this paper, we provide some results related to the $Δ_2$-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 $Δ_{2}$-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.

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