When fitting, by least squares, a linear model (with an intercept term) with $p$ parameters to $n$ data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided $p^3 \log^2 (p) /n \to 0$ for any error density having finite variance and a bounded first derivative. No further conditions are imposed on the sequence of design matrices. The result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares.
"Weak convergence of the empirical process of residuals in linear models with many parameters." Ann. Statist. 29 (3) 748 - 762, June 2001. https://doi.org/10.1214/aos/1009210688