The
autocovariance function is defined as the second moment product$$
\gamma (s,t)=cov(x_s,x_t)=E[(x_s- \mu_s)(x_t- \mu_t)]$$ for all \(s\),
\(t\). Note that for all time points s and t. The autocovariance
measures the linear dependence between two points on the same series
observed at different times. Recall from classical statistic \(\gamma
(s,t)=0\), \(x_t\) and \(x_s\) are not linearly related, but there
still may be some dependence structure between them. If, however,
\(x_t\) and \(x_s\) are bivariate normal, \(\gamma (s,t)=0\) ensures
their independence. It is clear that, for \(s = t\), the
autocovariance reduces to the (assumed finite) variance,$$\gamma
(s,t)=E[(x_s- \mu_s)(x_t- \mu_t)^2]=var(x_t).$$
Example:
The white noise series \(w_t\) has \(E(w_t)=0\) and $$ \gamma_w
(s,t)=cov(w_s,w_t)= \left\{
{\begin{array}{ccccccccccccccc}{\sigma^2_w,}&{{s} =
t,}\\{0,}&{{s} \ne t.}\end{array} } \right. $$