A limit theorem for sets of stochastic matrices

AbstractThe following fact about (row) stochastic matrices is an easy consequence of well known results: for each positive integer n⩾1 there is a positive integer q=q(n) with the property that if A is any n×n stochastic matrix then the sequence of matrices Aq,A2q,A3q,… converges. We prove a generalization of this for sets of stochastic matrices under the Hausdorff metric. Let d be any metric inducing the standard topology on the set of n×n real matrices. For a matrix A and set of matrices B define d(A,B) to be the infimum of d(A,B) over all B∈B. For two sets of matrices A and B, define d+(A,B) to be the supremum of d(A,B) over all A∈A, and define d(A,B) to be the maximum of d+(A,B) and d+(B,A). This is the Hausdorff metric on the set of subsets of n×n stochastic matrices. If A is a set of stochastic matrices and k is a positive integer, define A(k) to be the set of all matrices expressible as a product of a sequence of k matrices from A. We prove: For each positive integer n there is a positive integer p=p(n) such that if A is any subset of n×n stochastic matrices then the sequence of subsets A(p),A(2p),A(3p),… converges with respect to the Hausdorff metric