Wilcoxon-Signed-Rank Test

Abstract

This chapter introduces the basic ideas of the theory of weak convergence of a sequence of probability measures on complete and separable metric spaces, or equivalently, of convergence in distribution of a sequence of random values with values in such a space. The generality is not only of theoretical interest as it brings powerful and useful tools to bear on problems in the convergence of a sequence of real-valued stochastic processes. The development starts with a discussion of convergence in distribution for the simple case of a sequence of real-valued random variables and then moves on to the more complex cases. No proofs are given, but the results that are needed in the sequel are stated and discussed in the chapter. The ideas of weak convergence of probability measures find useful applications in many areas of probability and statistics as well as in many areas of application of those subjects, particularly in operations research and control theory. They frequently provide fundamental tools in the study of approximations. The approximations involve simpler processes—in particular, interpolated Markov chains—and the concepts of weak convergence theory is used to prove that the probability measures of the approximating sequences converge to the probability measure of the desired limit process and then to prove that various functionals of the simpler processes converge to the appropriate functionals of the limit. © 1977, Academic Press, Inc.