Convergence of Distributions in Metric Spaces

The subject of this chapter is the weak convergence of probability measures and the convergence in distribution of random variables that take on values in metric spaces. Keywords are the Portmanteau theorem, the mapping theorem, the subsequence criterion for weak convergence, Prokhorov’s theorem on tightness an relative compactness, and the notion of a convergence-determining class. In addition, various sufficient conditions for weak convergence are derived. A main example is the space C[0,1] of continuous functions defined on the unit interval. In this space, convergence in distribution follows from the convergence in distribution of all finite-dimensional distributions (fidi convergence) and relative compactness. According to the Arzel-Ascoli theorem, the latter condition may be stated in terms of the modulus of continuity. The chapter concludes with a result on convergence in distribution and independence, and with the notion of convergence in probability and a version of Slutsky’s lemma.