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.