3.3 Gaussian processes

Gaussian random processes

Gaussian process

A process with mean ${}_X(t)$ and covariance function $C_X(t_2, t_1)$ is said to be a Gaussian process if any $X=\left(\begin{array}{c}{X}_{t_1}\\ X_{t_2}\\ \\ X_{t_N}\end{array}\right)$ formed by any sampling of the process is a Gaussian random vector, that is,

$$f_X(x)=\frac{1}{(2\pi )^{\left(\frac{N}{2}\right)}\det {}_X^{\left(\frac{1}{2}\right)}}e^{-(\frac{1}{2}(x-_X)^T{}_X^{(-1)}(x-_X))}$$

for all $x\in \mathbb{R}^n$ where $$_X=\left(\begin{array}{c}{}_X(t_1)\\ \\ {}_X(t_N)\end{array}\right)$$ and $${}_X=\begin{pmatrix}{C}_X(t_1, t_1) & & C_X(t_1, t_N)\\ & \ddots \\ C_X(t_N, t_1) & & C_X(t_N, t_N)\\ \end{pmatrix}$$ .The complete statistical properties of $X_t$ can be obtained from the second-order statistics.

Properties

• If a Gaussian process is WSS, then it is strictly stationary.
• If two Gaussian processes are uncorrelated, then they are also statistically independent.
• Any linear processing of a Gaussian process results in a Gaussian process.