Abstract

Recent papers have shown that Πk = 1P(k) = limm→∞ (P(m) ... P(1)) exists whenever the sequence of stochastic matrices {P(k)}k = 1 exhibits convergence to an aperiodic matrix P with a single subchain (closed, irreducible set of states). We show how the limit matrix depends upon P(1).

In addition, we prove that limm→∞ limn→∞ (P(n + m) ... P(m + 1)) exists and equals the invariant probability matrix associated with P. The convergence rate is determined by the rate of convergence of {P(k)}k = 1 towards P.

Examples are given which show how these results break down in case the limiting matrix P has multiple subchains, with {P(k)}k = 1 approaching the latter at a less than geometric rate.

Authors
Format
Journal Article
Publication Date
Journal
Stochastic Processes and their Applications

Full Citation

. “The rate of convergence for backwards products of a convergent sequence of finite Markov matrices.”
Stochastic Processes and their Applications
vol.
11
, (May 01, 1981):
187
-
192
.