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

TL;DR

This paper explores the $ abla_2$-condition in Musielak-Orlicz functions and its implications for $\

Contribution

It establishes conditions under which $\\varphi$-families modeled on Musielak-Orlicz spaces are identical, and analyzes the boundary behavior of the normalizing function.

Findings

$\\varphi$-families are equal if modeled on Musielak-Orlicz spaces satisfying $\\Delta_2$-condition

Normalizing function behavior is characterized near the boundary of the $\\varphi$-family set

Results connect Musielak-Orlicz function properties with probability distribution families

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.