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 matrices H-n(+)(R) and more generally on an irreducible symmetric cone C. Then we study the convergence of random continued fractions on H-n(+)(R) and C by means of real Lagrangians for H-n(+)(R) and by new algebraic identities on symmetric cones for C. Finally we get a characterization of the inverse Gaussian distribution on H-n(+)(R) and C.