A good place to read about this is Adler and Taylor's book *Random Fields and Geometry*.   Regular random processes  or functions are  used more frequently in integral geometry.

 Consider  the more general  Gaussian  process  on $S^1$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\ii}{\boldsymbol{i}}$

$$  F(\theta)=\sum_{n\in\bZ} C_ne e^{n\ii \theta},  $$

where $C_n$  are independent  centered Gaussian random variables. Then $F$ is   a.s. smooth if $\newcommand{\vfi}{\varphi}$ $\newcommand{\bE}{\mathbb{E}}$  $\newcommand{\var}{\boldsymbol{var}}$ the  covariance kernel

$$K(\theta,\vfi)=\bE\bigl[ F(\theta)\overline{F(\vfi)}\bigr]= \sum_{n\in\bZ}\var[C_n] e^{\ii n(\theta-\vfi)} $$

is smooth. This happens if

$$\sum_{n\in\bZ} n^{2s}\var[C_n]<\infty,\;\;\forall s>0. $$


For stationary processes like this one this condition is also necessary.