Awi Federgruen

The rate of convergence for backwards products of a convergent sequence of finite Markov matrices

Abstract:

Recent papers have shown that Πk = 1 P(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.

Source: Stochastic Processes and their Applications
Exact Citation:
Federgruen, Awi. "The rate of convergence for backwards products of a convergent sequence of finite Markov matrices." Stochastic Processes and their Applications 11, no. 2 (May 1981): 187-192.
Volume: 11
Number: 2
Pages: 187-192
Date: 5 1981