📂Probability Theory

Gaussian Processes

Definition ¹

For all finite subsets $S = \left\{ X_{t_{k}} \right\}_{k=1}^{n} \subset \left\{ X_{t} \right\}$ of a stochastic process $\left\{ X_{t} \right\}$, if the linear combination $$ \sum_{k=1}^{n} a_{k} X_{t_{k}} \qquad , \left\{ a_{k} \right\}_{k=1}^{n} \subset \mathbb{R} $$ of elements of $S$ follows a multivariate normal distribution, then $\left\{ X_{t} \right\}$ is called a Gaussian Process.

Explanation

To non-experts, the definition might seem overly mathematical, but intuitively, it is not much different from the Wiener process. However, according to the definition, while the Wiener process is a Gaussian process, the reverse is not necessarily true.

Geometric Brownian motion follows a log-normal distribution at every point in time, so it is not a Gaussian process.

See also

• Gaussian Spatial Process