Multimodal convolutions of unimodal infinitely divisible distributions

For any positive integer $n$ an infinitely divisible distribution on $(0,\infty)$ such that its convolution with itself is $n$-modal is constructed. Moreover, we constructed the Lévy process $X_t$, $t\ge 0$, with the following properties: the distribution $X_t$ is not unimodal for $0<t<1$, is unimodal for $t=1$ and is $n$-modal for $t=2$.