A note on products of random matrices

Let P be a probability distribution on a set of d x d matrices. Let Pn denote the n-fold convolution of P with itself. Then tightness of the sequence Pn implies that the Cesáro sequence (1/n[sigma] Pn converges to an idempotent probability measure Q and the support of Q is exactly the set m(S) of matrices of minimal rank in the semigroup S generated by the support of P. Furthermore the set m(S) is a completely simple semigroup with compact group factor. The convergence of Pn can be characterized in terms of the Rees-Suschkewitsch decomposition of m(S).