Gaussian process play an important role in random signal processing. A gaussian process is simply a random signal $X$ such that at each time $t$, $X(t)$ is a gaussian random variable. We give here a short reminder on gaussian random variables.
Real gaussian random variable\
Let $X$ be a random variable. $X$ is a real gaussian random variable iff its probability density function reads
$$
p_X(u) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(u-\mu)^2}{2\sigma^2}}
$$
where $\mu$ is the expectation of $X$, and $\sigma^2$ its variance.
Real gaussian random vector\
Let $\boldsymbol{X}$ be a random vector. $\boldsymbol{X}$ is a real gaussian random vector iff every linear combination of its components is a real random variable.
Let $\boldsymbol{X}$ be a gaussian random vector of size $M$, let $\boldsymbol{\mu}\in\mathbb{R}^M$ its expectation and $\boldsymbol{\Sigma}\in\mathbb{R}^{M\times M}$ its covariance matrix: $$ \boldsymbol{\mu} = \text{E}(\boldsymbol{X})\quad \boldsymbol{\Sigma}=\text{E}\left((\boldsymbol{X}-\boldsymbol{\mu})(\boldsymbol{X}-\boldsymbol{\mu})^T\right) $$ We denote by $\boldsymbol{X} \sim \mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right). If $\boldsymbol{\Sigma}$ is invertible, its probability density function reads $$ p_{\boldsymbol{X}}(\boldsymbol{u}) = \frac{1}{(2\pi)^{\frac{M}{2}}\sqrt{|\text{det}\left(\boldsymbol{\Sigma}\right)|}}e^{-\frac{1}{2}(\boldsymbol{u}-\boldsymbol{\mu})^T\boldsymbol{\Sigma}^{-1}(\boldsymbol{u}-\boldsymbol{\mu})} $$
Real Gaussian process\
Let $X$ be a real stochastic process. $X$ is said Gaussian iff for all finite set $\lbrace t_1,\ldots,t_m \rbrace of instants, the random vector $\boldsymbol{X} = (X(t_1),\ldots,X(t_m))$ is a real gaussian vector.
If the random variable is complex, the circularity means the invariance by rotation in the complex plan of the statistics. In other words, for all $\phi\in\mathbb{R}$, $X$ and $Xe^{i\phi}$ have the same probability density function. Then, circular complex random variable are of zero mean. One has $$ \text{E}(X) = 0 \quad \text{ and } \quad \text{E}(X^2) = 0 $$ Remark for a zero mean complex random variable $\text{E}(X^2)$ is not the variance of $X$. Indeed, the variance is given by $\text{Var}(X) = \text{E}(X\overline{X})$.
Complex circular gaussian random variable\
Let $X$ be a random variable. $X$ is a complex circular gaussian random variable iff its probability density function reads
$$
p_X(u) = \frac{1}{\sigma\pi}e^{-\frac{|u|^2}{\sigma^2}}
$$
where $\sigma^2$ is the variance of $X$.
Complex circular gaussian random vector\
Let $\boldsymbol{X}$ be a random vector. $\boldsymbol{X}$ is a complex circular gaussian random vector iff every linear combination of its components is a complex circular random variable.
Let $\boldsymbol{X}$ be a complex circular gaussian random vector of size $M$, let $\boldsymbol{\Sigma}\in\mathbb{R}^{M\times M}$ its covariance matrix: $$ \boldsymbol{\Sigma}=\text{E}\left(\boldsymbol{X}\boldsymbol{X}^{*}\right) $$ We denote by $\boldsymbol{X}\sim\mathcal{CN}\left(\boldsymbol{0},\boldsymbol{\Sigma}\right)$. If $\boldsymbol{\Sigma}$ is invertible, its probability density function reads $$ p_{\boldsymbol{X}}(\boldsymbol{u}) = \frac{1}{\pi^{M}|\text{det}\left(\boldsymbol{\Sigma}\right)|} e^{-\boldsymbol{u}^{*}\boldsymbol{\Sigma}^{-1}\boldsymbol{u}} $$
Complex circular Gaussian process\
Let $X$ be a complex stochastic process. $X$ is said complex circular Gaussian iff for all finite set $\lbrace t_1,\ldots,t_m \rbrace of instants, the random vector $\boldsymbol{X} = (X(t_1),\ldots,X(t_m))$ is a complex circular gaussian vector.
A Gaussian process being perfectly determined by its first two moments, the weak stationarity implies the strong stationarity.