A theorem for random Fourier series on compact quantum groups

Helgason showed that a given measure $f\in M(G)$ on a compact group $G$ should be in $L^2(G)$ automatically if all random Fourier series of $f$ are in $M(G)$. We explore a natural analogue of the theorem in the framework of compact quantum groups and apply the obtained results to study complete representability problem for convolution algebras of compact quantum groups as an operator algebra.