We show that, for any stochastic event p of period n, there exists a
measure-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 an
algorithm which constructs the desired 1qfa in polynomial time. As a consequence, we get that any periodic language of period n can be
accepted 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-once
1qfa's with respect to classical automata.