Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH_n⁺() and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH_n⁺() andC by means of real Lagrangians forH_n⁺() and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH_n⁺() andC.