 Covariance criterion for diversification of factor residuals

For centered residual random variables ${ϵ}_i$ 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 $\mathrm{Var}(w^Tϵ)\le s^2{W}^2/n$, hence convergence to zero in mean-square convergence. For nonnegative weights summing to one, the sharper bound is $s^{2}W/n$. 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.