  We compute the ${\mathbb F}_p$-dimension of an $n$-th graded piece
$G_{(n)}/G_{(n+1)}$ of the Zassenhaus filtration for various finitely generated
pro-$p$-groups $G$. These groups include finitely generated free
pro-$p$-groups, Demushkin pro-$p$-groups and their free pro-$p$ products. We
provide a unifying principle for deriving these dimensions.
