The Annals of Mathematical Statistics

Realization of Stochastic Systems

Michael Arbib

Full-text: Open access


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

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

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



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

Export citation