19.3. Special case: Gaussian process#
The joint density function for a multivariate normal distribution was presented in Eq. (37.33) for \(\boldsymbol{x} = (x_1, x_2, \ldots, x_k)\) corresponding to random variables \(X_1, X_2, \ldots X_k\). This normal distribution is completely determined by its mean vector, \(\boldsymbol{\mu}\), and covariance matrix, \(\boldsymbol{\Sigma}\), where elements \(\mu_i = \expect{X_i}\) and \(\Sigma_{ij} = \cov{X_i}{X_j}\).
The multivariate normal distribution has the remarkable property that all marginal and conditional distributions are normal, and specified by the corresponding subsets of the mean vector and covariance matrix.
A Gaussian process extends the multivariate normal distribution to a stochastic process with a continuous time index \(T\).
Definition 19.1 (Gaussian process)
A Gaussian process \((X(t))_{t \geq 0}\) is a stochastic process with a continuous-time index \(t \in [0,\infty)\) if each finite-dimensional vector of random variables
has a multivariate normal distribution for \(0 \leq t_1 < \ldots < t_k\).
A Gaussian process is completely determined by its mean function \(\mu(X(t))\) and covariance function \(C(X(s), X(t))\) for \(s, t \geq 0\).
A Gaussian process is stationary if the mean function \(\mu(X(t))\) is constant for all \(t\) and if the covariance function fulfils \(C(X(s), X(t)) = C(X(s+h), X(t+h))\) with \(h \geq 0\). Note that stationarity is not a requirement for a Gaussian process.
We return to discuss Gaussian processes in detail in Overview of Gaussian processes.