= Finite-dimensional Brownian bridge limit of an empirical distribution process
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For fixed $x_1,\ldots,x_d$, 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.
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