Infinite products of substochastic matrices
Norman J. Pullman · Pacific Journal of Mathematics · 1966
This paper is about two types of infinite products of substochastic matrices {Aj} namely: the left product defined by the sequence of left partial products A 1$ A 2 A ί9 A Z A 2 A U •••; and the right product defined by the sequence of right partial products A ίf A X A 2 , AιA 2 A z ,The basic theorem is that if the A n are each oo by oo then: a.There is a nonempty set E of substochastic sequences each of which (except possibly the zero sequence, 0) is the componentwise limit of a sequence of rows, one from each left partial product; b.Any sequence {ρ n } of rows, one from each left partial product, can be approximated by a sequence of convex combinations {c n } of points of E (that is, {p n -c n } converges componentwise to the zero sequence), and c.E = {0} if and only if every sequence of rows, one from each left partial product, converges to 0.Similar conclusions follow immediately for the right product of oo by oo doubly substochastic matrices.The asymptotic behaviour of the right product of a special class of {A w } is also considered.