## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 3 (1974), 530-534.

### On the Approximation of Stationary Measures by Periodic and Ergodic Measures

#### Abstract

Let $(\Omega, \mathscr{F})$ be the measurable space consisting of $\Omega$, the set of sequences $(x_1, x_2, \cdots)$ from a finite set $A$, and $\mathscr{F}$, the usual product sigma-field. Let $X_1, X_2, \cdots$ be the usual coordinate random variables defined on $\Omega$. For $n = 1,2, \cdots$, let $\mathscr{F}_n$ be the sub sigma-field of $\mathscr{F}$ generated by $X_1, X_2, \cdots, X_n$. We prove the following: if $P$ is a probability measure on $\mathscr{F}$ stationary with respect to the one-sided shift transformation on $\Omega$ and if $N$ is a positive integer, then there is a periodic measure $Q$ on $\mathscr{F}$ such that $Q = P$ over $\mathscr{F}_N$. This is a stronger result than the known fact that the periodic measures are dense in the set of stationary measures under the weak topology. We also show that if $P$ assigns positive measure to every non-empty set in $\mathscr{F}_N$, it is possible to find an ergodic measure $Q$ such that $P = Q$ over $\mathscr{F}_N$. We investigate the entropies of all such ergodic measures $Q$ which approximate $P$ in this sense, and show that there is a unique ergodic measure $Q$ of maximal entropy such that $P = Q$ over $\mathscr{F}_N$.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 3 (1974), 530-534.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996671

**Digital Object Identifier**

doi:10.1214/aop/1176996671

**Mathematical Reviews number (MathSciNet)**

MR369661

**Zentralblatt MATH identifier**

0287.60033

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B05: Probability measures on topological spaces

Secondary: 94A15: Information theory, general [See also 62B10, 81P94]

**Keywords**

28-A65 28-A35 Stationary measures ergodic measures periodic measures entropy shift transformation

#### Citation

Kieffer, J. C. On the Approximation of Stationary Measures by Periodic and Ergodic Measures. Ann. Probab. 2 (1974), no. 3, 530--534. doi:10.1214/aop/1176996671. https://projecteuclid.org/euclid.aop/1176996671