Convergence in Distribution—Functional Results

This chapter addresses functional central limit theorems, that is invariance principles and the convergence of empirical processes. The importance of these processes come, of course, from the several statistical applications that are based on transformations of the random-sum process or of the empirical process. Both these sequences of processes are shown to converge in distribution to suitable Gaussian processes. Some transforms depend closely on the paths of processes, while others are only integral transformations, thus being less sensitive to the regularity of the observed path. These arguments justify that, depending on the functionals that we are interested in, we may require the convergence with respect to the usual Skorokhod space or with respect to some suitable L p space. These, being weaker topological spaces, will be less demanding in order to have the convergence in distribution. The techniques are similar to those used in Chap. 4 but adapted to handle the technicalities that arise from the underlying functional space.