Convergence in distribution of nonmeasurable random elements

Abstract

A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jørgensen, is characterized in terms of weak convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit. In the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.