Riemannian Holonomy Groups of Statistical Manifolds

Normal distribution manifolds play essential roles in the theory of information geometry, so do holonomy groups in classification of Riemannian manifolds. After some necessary preliminaries on information geometry and holonomy groups, it is presented that the corresponding Riemannian holonomy group of the $d$-dimensional normal distribution is $SO\left(\frac{d\left(d+3\right)}{2}\right)$, for all $d\in\mathbb{N}$. As a generalization on exponential family, a list of holonomy groups follows.Comment: 11 page

Projection Solutions of Frobenius-Perron Operator Equations

We construct in this paper the first order and second order piecewise polynomial finite approximation schemes for the computation of invariant measures of a class of nonsingular measurable transformations on the unit interval of the real axis. These schemes are based on the Galerkin’s projection method for L1-spaces and are proved to be convergent for the class of Frobenius-Perron operators