# 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)* = *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.

## 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.

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