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.