= Adjacent products of independent Gaussian variables
{title2=$\epsilon_1=\eta_1,\quad\epsilon_i=\eta_{i-1}\eta_i$}
For independent <standard normal random variables> $\eta_1,\ldots,\eta_n$, define the displayed errors for $i\ge2$. Each has <mean> zero and <variance> one. Nonadjacent errors depend on disjoint sets of independent variables. Adjacent <covariances> vanish because their product contains an independent centered factor. Thus their <covariance matrix> is $I_n$.
They are nevertheless dependent: the <Gaussian fourth moment> gives $\mathbb E[\epsilon_i^2\epsilon_{i+1}^2]=3$, and hence $\operatorname{Cov}(\epsilon_i^2,\epsilon_{i+1}^2)=2$. Constant <variance> and zero pairwise <covariance> can conceal dependence in magnitudes. This is useful when interpreting a <residual plot> in <linear regression>.
Back to article page
