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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.
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