  We introduce the Gaussian part of a compact quantum group $\mathbb{G}$,
namely the largest quantum subgroup of $\mathbb{G}$ supporting all the Gaussian
functionals of $\mathbb{G}$. We prove that the Gaussian part is always
contained in the Kac part, and characterise Gaussian parts of classical compact
groups, duals of classical discrete groups and $q$-deformations of compact Lie
groups. The notion turns out to be related to a new concept of "strong
connectedness" and we exhibit several examples of both strongly connected and
totally strongly disconnected compact quantum groups.
