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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.

A residual plot displays regression residuals against fitted responses, predictor values or observation order. Curvature can suggest an incorrect regression function; changing spread can suggest unequal error variances; ordered runs can suggest dependence. A plot is a diagnostic, not a proof of independence. Dependence confined to higher moments, as for adjacent products of independent Gaussian variables, may call for lagged squared regression residuals rather than a basic scatter plot. Fitting also induces dependence among regression residuals even under independent original errors.