On a Converse to Scheffe's Theorem

Abstract

In Boos (1985) equicontinuity conditions are given which ensure the uniform convergence of densities in $\mathbb{R}^k$, given convergence in distribution. In the present note we show that such equicontinuity conditions in fact characterize uniform local convergence with no additional assumptions on the sequence of densities, or on the limit density. Versions of these results are also given when the distributions depend on an unknown parameter; these forms will be relevant for the uniform approximation of likelihood functions.