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This paper concerns the locally uniform weak convergence of functions takingvalues in the space of finite Borel measures on a metric space. In particular,we establish a locally uniform version of L\'evy's continuity theorem forcharacteristic functions, assuming that the metric space is a Euclid space. Asapplication, we show that the locally uniform weak convergence of the marginaldistributions of additive processes characterizes the functional convergence ofthe processes.

Notes on locally uniform weak convergence with application to additive processes