We show that, for any stochastic event p of period n, there exists ameasure-once one-way quantum finite automaton (1qfa) with at most $2\sqrt{6n}+25$
states inducing the event ap+b, for constants a>0, b ≥ 0, satisfying a+b ≥ 1. This fact is proved by designing analgorithm which constructs the desired 1qfa in polynomial time. As a consequence, we get that any periodic language of period n can beaccepted with isolated cut point by a 1qfa with no more than $2\sqrt{6n}+26$
states. Our results give added evidence of the strength of measure-once1qfa's with respect to classical automata.