کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4615125 | 1339308 | 2015 | 19 صفحه PDF | دانلود رایگان |

Given a probability measure μ , on the space of strictly positive densities MμMμ, we construct a topological manifold on which the elements are connected by κ -exponential models in the form q=expκ(u⊖κKp,κ(u))p, where expκ(x)=(1+κ2x2+κx)1/κ, x⊖κy=x1+κ2y2−y1+κ2x2, p,q∈Mμp,q∈Mμ, and their local representations are elements of an Orlicz space, i.e. the manifold is modeled on Orlicz spaces. Parameter k is the G. Kaniadakis parameter for κ -deformed exponentials which is strongly relevant to relativity and statistical complex models in statistical mechanics. Functional Kp,κKp,κ is the deformed counterpart of the cumulant mapping and satisfies that, if κ→0κ→0, we obtain the usual cumulant functional of the exponential manifold; moreover in this limit case the exponential manifold constructed by Pistone and Sempi is recovered. In the context of deformed exponentials, we prove that the function ϕκ(⋅)=coshκ(⋅)−1ϕκ(⋅)=coshκ(⋅)−1, where coshκ(x)coshκ(x) is the κ-deformed hyperbolic cosine, is a Young function and generates the Orlicz space on which the κ -exponential manifold is modeled, namely Lϕκ(p⋅μ)Lϕκ(p⋅μ). This construction differs from the one made by Pistone on the paper κ-exponential models from the geometrical viewpoint , since this last one is based on divergence functionals and modeled on Lebesgue spaces L1/κ(p⋅μ)L1/κ(p⋅μ). The use of κ-deformed models is interesting since they generalize the exponential models and extend them to non-additive systems.
Journal: Journal of Mathematical Analysis and Applications - Volume 431, Issue 2, 15 November 2015, Pages 1080–1098