Abstract
In this paper, we provide some results related to the Δ2-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 Δ2-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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References
Kamińska, A.: Some convexity properties of Musielak-Orlicz spaces of Bochner type. In: Proceedings of the 13th Winter School on Abstract Analysis (Srní, 1985), vol. 10, pp. 63–73 (1986)
Kaniadakis, G.: Statistical mechanics in the context of special relativity. Phys. Rev. E (3), 66(5), 056125, 17 (2002)
Kolwicz, P., Płuciennik, R.: On P-convex Musielak-Orlicz spaces. Comment. Math. Univ. Carolin. 36(4), 655–672 (1995)
Krasnoselśkiĭ, M.A., Rutickiĭ, J.: Convex functions and Orlicz spaces. Translated from the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen (1961)
Lang, S.: Differential and Riemannian manifolds, 3rd edn. Graduate Texts in Mathematics, vol. 160. Springer, New York (1995)
Musielak, J.: Orlicz spaces and modular spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
Pistone, G., Rogantin, M.P.: The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli 5(4), 721–760 (1999)
Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statist. 23(5), 1543–1561 (1995)
Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, p. 449. Marcel Dekker Inc., New York (1991)
Vigelis, R.F., Cavalcante, C.C.: On ϕ-families of probability distributions. J. Theor. Probab., 1–15 (2011) (article in press), doi:10.1007/s10959-011-0400-5
Vigelis, R.F., Cavalcante, C.C.: Smoothness in Musielak–Orlicz function spaces equipped with the Orlicz norm (2012) (submitted for publication)
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Vigelis, R.F., Cavalcante, C.C. (2013). The Δ2-Condition and ϕ-Families of Probability Distributions. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_81
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DOI: https://doi.org/10.1007/978-3-642-40020-9_81
Publisher Name: Springer, Berlin, Heidelberg
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