Vol. 16, No. 3, 1966

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Infinite products of substochastic matrices

Norman Jay Pullman

Vol. 16 (1966), No. 3, 537–544

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 A1,A2A1,A3A2A1, ; and the right product defined by the sequence of right partial products A1,A1A2,A1A2A3, .

The basic theorem is that if the An are each by 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 {ρn} of rows, one from each left partial product, can be approximated by a sequence of convex combinations {cn} of points of E (that is, {ρn cn} 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 by doubly substochastic matrices.

The asymptotic behaviour of the right product of a special class of {An} is also considered.

Mathematical Subject Classification
Primary: 60.65
Received: 9 October 1964
Published: 1 March 1966
Norman Jay Pullman