Empirical Processes: General Weak Convergence Theory

The lack of measurability of the empirical process with respect to the sigma-field generated by the ‘natural’ l∞ metric, as illustrated in the previous notes, needs an extension of the standard weak convergence theory that can handle situations where the converging stochastic processes may no longer be measurable (though the limit will be a tight Borel measurable random element). Of course, an alternative solution is to use a different metric that will make the converging processes measurable, which in the scenario of Donsker’s theorem, Version 1 in the previous notes is achieved by equipping the space D[0, 1] with the Skorohod (metric) topology. However, for empirical processes that assume values in very general (and often more complex spaces) such cute generalizations are not readily achievable and the easier way out is to keep the topology simple and tackle the measurability issues. Of course, any generalization of the notion of weak convergence must allow a powerful continuous mapping theorem.