2.9 Gaussian Processes

We will often deal with Gaussian temporal processes which are stationary processes whose joint distribution is Gaussian. That is, for the series \(Y_1,\dots,Y_n\), they are jointly distributed as \(\mathcal{N}(\mathbf{1}\mu, \Sigma(\gamma))\), where \(\gamma\) is the autocovariance function such that \(\gamma(k) = \text{Cov}(Y_j, Y_{j+k})\) for all integers \(j\) and \(k\). Note that we also have \(\gamma(0) = \text{Cov}(Y_j, Y_j) = \text{Var}(Y_j)\) for all \(j\).

From the Cauchy-Schwarz inequality, we can see that \[\begin{eqnarray*} \text{Cov}(Y_j,Y_{j+k})^2 & \leq & \text{Var}(Y_j)\text{Var}(Y_{j+k})\\ & = & \gamma(0)^2\\ |\text{Cov}(Y_j,Y_{j+k})| & \leq & \gamma(0) \end{eqnarray*}\]

In practice, we generally assume \(\gamma(k)\rightarrow 0\) as \(k\rightarrow\infty\) so that the dependence between observations \(Y_j\) and \(Y_{j+k}\) decays after a certain lag distance \(k\). If it is the case that \(\gamma(k)=0\) for \(k > m\), then the time series is referred to as an \(m\)-dependent series.