The Space \({\mathrm {D}} [0,1]\) , Empirical Processes

This chapter deals with convergence in distribution the Càdlàg space \(\mathrm {D} := \mathrm {D}[0, 1]\) of all right-continuous real-valued functions defined on the unit interval \([0, 1]\) , whose left-hand limits exist at every point \(t > 0\) . Topics are the Skorokhod metric and criteria for the convergence in distribution for D-valued random elements, although not all technical details are given. The main results are Donsker’s theorem in the space D and the convergence in distribution of the empirical process. The first application is the limit null distribution the test statistic of the celebrated Kolmogorov goodness-of-fit test. This limit distribution is the distribution of the maximum modulus of the Brownian bridge. The same limit distribution shows up under the hypothesis of equality of distributions for the test statistic of Kolmogorov and Smirnov in the context of the two-sample problem. The chapter also considers the Cramér-von Mises statistic when testing for uniformity on the unit interval. Here, the limit null distribution is the distribution of the integrated square of a Brownian bridge.