## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 38, Number 3 (1967), 927-933.

### Realization of Stochastic Systems

#### Abstract

Heller has given necessary and sufficient conditions that a stochastic process be induced from a Markov chain. We consider a process induced by a Markov chain to be a probabilistic finite automaton with one input. With each state of a probabilistic finite automaton, we may associate a function $p(u \mid v)$, which tells us the probability that, if we apply the input sequence $v$ to the machine started in the state, we should observe output sequence $u$. We give a necessary and sufficient condition that a function $p(u \mid v)$ be realizable as much as input-output function. Finally, we show Heller's result is extended by our condition.

#### Article information

**Source**

Ann. Math. Statist., Volume 38, Number 3 (1967), 927-933.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177698889

**Digital Object Identifier**

doi:10.1214/aoms/1177698889

**Mathematical Reviews number (MathSciNet)**

MR225606

**Zentralblatt MATH identifier**

0147.36801

**JSTOR**

links.jstor.org

#### Citation

Arbib, Michael. Realization of Stochastic Systems. Ann. Math. Statist. 38 (1967), no. 3, 927--933. doi:10.1214/aoms/1177698889. https://projecteuclid.org/euclid.aoms/1177698889