A limit theorem for sets of stochastic matrices Original Research Article

The following fact about (row) stochastic matrices is an easy consequence of well known results: for each positive integer ngreater-or-equal, slanted1 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 image define image to be the infimum of d(A,B) over all image. For two sets of matrices image and image, define image to be the supremum of image over all image, and define image to be the maximum of image and image. This is the Hausdorff metric on the set of subsets of n×n stochastic matrices. If image is a set of stochastic matrices and k is a positive integer, define image to be the set of all matrices expressible as a product of a sequence of k matrices from image. We prove: For each positive integer n there is a positive integer p=p(n) such that if image is any subset of n×n stochastic matrices then the sequence of subsets image converges with respect to the Hausdorff metric.