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.

Let $x: \Omega \to (A \to \R )$ be a normal
random function with family $X: A \to (\Omega
\to \R )$.
There exists unique functions $m: A \to \R $
and $k: A \times A \to \R $ so that the mean
of the random variable $X_a$ is $m(a)$ for all
$A$ and the covariance of the random variables
$X_a$ and $X_{a'}$ is $k(a, a')$ for
all $a, a' \in A$.^{2}

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

\[ \pmat{ k(a^1, a^1) & \cdots & k(a^1, a^m) \\ \vdots & \ddots & \vdots \\ k(a^m, a^1) & \cdots & k(a^m, a^m) \\ } \]

is positive semidefinite, then we can construct a Gaussian process with mean function $m$ and covariance function $k$.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.

- Future editions will expand. ↩︎
- Future editions may include an account. ↩︎
- Some authors belabor this point because of the
natural inclination to want to specify an
*inverse*covariance function, which need not satisfy the consistency property. The consistency property ensures that any marginal of a subfamilys density is the density of that further subfamily. Future editions may expand. ↩︎