A Discussion of Weak Convergence
by
John Fresen
Department of Statistics, University of Missouri

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, a number of processes display properties of weak convergence despite being non-measurable. They do not obey the formal definition, yet they converge ‘in spirit’. The first such example was the uniform empirical process (Chibisov 1965, Billingsley 1968), which converges to a Brownian bridge.

In this talk, we discuss the formulation of weak convergence of random elements in an abstract space and mention a number of attempts to overcome non-measurability. In the spirit of this conference, we highlight an obvious but not universally known connection to weak-star convergence in Banach space.

Date received: April 14, 2008