A Theorem in the Theory of Infinitely Divisible Laws

Let $\mathfrak{F}$ be a class of infinitely divisible distribution functions $F$ for which, if $F\in\mathfrak{F}$ and $F*H=Q$, where $Q(x)$ is an infinitely divisible distribution function, it follows that $H$ is also an infinitely divisible distribution function. The following theorem is proved:
Class $\mathfrak{F}$ is identical to the set of all normal distributions.