Finite-dimensional Brownian bridge limit of an empirical distribution process

For fixed $x_1,\dots ,x_d$, the vector $({\alpha }_n(x_j){)}_{j=1}^d$ converges in distribution to $(B(G(x_j)){)}_{j=1}^d$, where $B$ is a standard Brownian bridge. To prove it, write the vector as $n^{-1/2}$ times the sum of independent centred bounded indicator vectors. Their covariance matrix is $G(\min (x_i,x_j))-G(x_i)G(x_j)$. The multivariate central limit theorem gives a centred multivariate normal distribution with this covariance, exactly the bridge vector. Singular covariance matrices, including endpoint times or repeated values of $G$, cause no problem. This is a finite-dimensional assertion; a full function-space limit additionally requires tightness.