Normalization and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}-function: Definition and Consequences

• Published in: Geometric Science of Information pp. 231 - 238
• Main Authors: de Andrade, Luiza H. F., Vigelis, Rui F., Vieira, Francisca L. J., Cavalcante, Charles C.
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Exponential Family; Infinite Dimensional Setting; Measurable Function; Musielak Orlicz; Preliminary Considerations
• ISBN: 9783319684444; 3319684442
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-68445-1_27

It is known from the literature that a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}-function may be used to construct the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}-families of probability distributions. In this paper, we assume that one of the properties in the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}-function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}-function affects the behavior of the normalizing function.