Analytical and geometrical properties of statistical connections in Information Geometry

Information Geometry is a field where one can measure the deep impact of geometry and analysis in statistics, information theory and related applied fields. The present contribution has the goal of showing also the impact that statistics and information theory can have in geometry and analysis. Indeed it is clear that the development of the non-parametric and non-commutative versions of Information Geometry need a massive use of mathematical instruments of infinite-dimensional analysis, geometry and operator theory. On the other side there is an increasing interest of mathematicians for the non-trivial problems that are suggested by the application of Information Geometry. Even in the elementary case of a finite state space, the geometrical approach adds considerable insight to the modeling of applied problems.