Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 7 / c / ii / Solution

Independence of the centered standard normal random variables gives $\mathbb{E}{ϵ}_i=0$ for every $i$. AlsoFor nonadjacent indices, the underlying sets of $\eta$ variables are disjoint, so the errors are independent and have zero covariance. For adjacent indices with $i\ge 2$,The first adjacent pair separately gives $\mathbb{E}[{ϵ}_1{ϵ}_2]=\mathbb{E}[{\eta }_1^2{\eta }_2]=0$. HenceThese adjacent products of independent Gaussian variables are uncorrelated random variables even though adjacent errors share a factor and are dependent.