functions on [0,1], such that convergence for d to a continuous function is equivalent to convergence for the sup norm, and showed that Gn converges in law in to the Brownian bridge. Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove that there exist Xi, iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges Bn, such that is measurable and converges in probability to 0. An improved version of this result, providing more detail on the rate of convergence, is the Komlós–Major–Tusnády approximation. See also Glivenko–Cantelli theorem Kolmogorov–Smirnov test References Probability theorems Theorems in statistics Empirical process