A uniform renewal theorem

It is shown that the Renewal Theorem holds uniformly with respect to the step distribution F when the latter is restricted a weakly closed class K of non-arithmetic distribution functions for which |x| is uniformly integrable with respect to F and the means pF are bounded away from 0. The proof uses a coupling argument of which a uniform version of-the Recurrence Theorem is an interesting byproduct. An application to asymptotic local minimaxity in sequential estimation is described.