We want to discuss real-valued random functions whose family of random variables have simple densities.1
A normal random function is a real-valued random function whose family of real-valued random variables has the property that any subfamily is jointly normal.
For this reason, we call the family of random variables (or stochastic process) corresponding to the random function a gaussian process or normal process.
Let $(\Omega , \mathcal{A} , \mathbfsf{P} )$ be a probability space and $A$ a set. Let $x: \Omega \to (A \to \R )$ be a random function with family $y: A \to (\Omega \to \R )$.
The random function $x$ is a normal if, for all $a^1, \dots , a^m \in A$, $(y(a^1), \dots , y(a^m))$ is jointly normal.
For this reason, we call $m$ the mean function and $k$ the covariance function of the random function.
Conversely, let $m: A \to \R $ and $k: A
\times A \to \R $.
Then if $k$ satisfies the property that for
all $a^1, \dots , a^m$, the $m \times m$ matrix
\[
\pmat{
k(a^1, a^1) & \cdots & k(a^1, a^m) \\
\vdots & \ddots & \vdots \\
k(a^m, a^1) & \cdots & k(a^m, a^m) \\
}
\]
Let $A = \set{1, \dots , n}$ and let $K \in \R ^{n \times n}$ be symmetric positive semidefinite. Define $m: A \to \R $ to be $m \equiv 0$ (the constant zero function) and $k(i, j) = K_{ij}$. Then the normal random function $x: \Omega \to (A \to \R )$ with mean $m$ and covariance $k$ is in one to one correspondence with the gaussian random vectors with mean zero.