## Awi Federgruen

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

**Abstract:**

Recent papers have shown that Π^{∞}_{k = 1} *P(k)* = *lim*_{m}→∞ (*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 *lim*_{m}→∞ *lim*_{n}→∞ (*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