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.

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 $S^1$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\ii}{\boldsymbol{i}}$

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

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.

A good place to read about this is Adler and Taylor's book Random Fields and 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.

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.