A Matsumoto–Yor property for Kummer and Wishart random matrices

International audienceFor a positive integer r, let I denote the r × r unit matrix. Let X and Y be two independent r × r real symmetric and positive definite random matrices. Assume that X follows a Kummer distribution while Y follows a non-degenerate Wishart distribution, with suitable parameters. This note points out the following observation: the random matrices U := [I + (X + Y) −1 ] 1/2 [I + X −1 ] −1 [I + (X + Y) −1 ] 1/2 and V := X + Y are independent and U follows a matrix beta distribution while V follows a Kummer distribution. This generalizes to the matrix case an independence property established in Koudou and Vallois (2010) for r = 1