= Covariance criterion for diversification of factor residuals
{title2=$\operatorname{Var}(w^T\epsilon)=w^T\Gamma w$}
For centered residual <random variables> $\epsilon_i$ with <covariance matrix> $\Gamma$, the residual of a <portfolio> has <variance> $w^T\Gamma w$. Pairwise uncorrelated residuals with uniformly bounded <variances> and weights $|w_i|\le W/n$ give $\operatorname{Var}(w^T\epsilon)\le s^2W^2/n$, hence convergence to zero in <mean-square convergence>. For nonnegative weights summing to one, the sharper bound is $s^2W/n$. With correlated residuals, $|w_i|\le W/n$ and $\sum_{i,j}|\Gamma_{ij}|=o(n^2)$ also suffice. Individual bounded <variances> alone do not: if every residual equals the same nondegenerate centered $Z$, every fully invested <portfolio> retains residual $Z$. A one-sided upper bound on signed weights is insufficient even for uncorrelated residuals because it allows a fixed large <short selling>.
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