Weak convergence of probability measures relative to incompatible topology and ?-field with applications to renewal theory

Abstract

Necessary and sufficient conditions are given in order that a sequence of probability measures, weakly convergent relative to a given topology τ 0 and associated σ-field σ(τ 0), are weakly convergent (and satisfy a continuity theorem) relative to the σ(τ 0)-measurable functions which are continuous in some finer topology τ 1, even if μ does not extend to σ(τ 0). These conditions are shown to be applicable to a sequence of translated renewal measures. Alternate conditions (tightness, uniformity of weak convergence) are investigated and shown to be inappropriate.