Generic Uniform Convergence

Abstract

This paper presents several generic uniform convergence results that include generic uniform laws of large numbers. These results provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence. The results are useful for establishing asymptotic properties of estimators and test statistics.The results given here have the following attributes, (1) they extendresults of Newey to cover convergence almost surely as well as convergence in probability, (2) they apply to totally bounded parameter spaces (rather than just to compact parameter spaces), (3) they introduce a set of conditions for a generic uniform law of large numbers that has the attribute of giving the weakest conditions available for i.i.d. contexts, but which apply in some dependent nonidentically distributed contexts as well, and (4) they incorporate and extend themain results in the literature in a parsimonious fashion.