On the Definition of Weak Convergence of a Sequence of Random Elements
Department of Statistics
University of Missouri-Columbia
The classical formulation of weak convergence of a sequence of random elements is based intrinsically on the assumption that each of the random elements in the sequence is Borel measurable. However, Chibisov (1965) and Billingsley (1968) pointed out that the members of the sequence of the uniform empirical processes are not Borel measurable but nonetheless display some properties of weak convergence to a Brownian Motion. Thus the definition of weak convergence seems inadequate in that we have a sequence that does not satisfy the classical formulation yet it has some properties similar to weak convergence.This is joint work with Daniel Fresen.
Subsequently, many other examples of a sequence of non-measurable functions that display similar properties to those of weak convergence have been discovered in the stochastic process literature. Over the last three decades a number of attempts have been proposed to modify or generalize the definition of weak convergence to overcome this apparent anomalous inadequacy. However the solution, as yet, has proved illusive.
This paper summarizes the classical formulation of weak convergence of a sequence of random elements in an abstract space. We then look at examples of sequences of non-measurable functions that display properties similar to weak convergence. We summarize some of the proposed modifications and generalizations of weak convergence to overcome this problem. We focus mainly on the work of Skorokhod, Billingsley, Dudley, Pike, Shorack, Wellner, Hoffmann-Jorgensen and van der Vaart. We discuss some of the difficulties with these proposals and make suggestions.