Irregular Behaviours for Probabilistic Automata

We consider probabilistic automata over finite words. Such an automaton defines the language consisting of the set of words accepted with probability greater than a given threshold. We show the existence of a universally non-regular probabilistic automaton, i.e. an automaton such that the language it defines is non-regular for every threshold. As a corollary, we obtain an alternative and very simple proof of the undecidability of determining whether such a language is regular.