Abstract
For a locally Lipschitz continuous mappingF between metric spacesZ andX and probability measures μ, ν onZ, bounds for the (Prokhorov and bounded Lipschitz) distance of μF −1 and νF −1 are obtained in terms of the distance of μ and ν, the growth of the local Lipschitz constants ofF, and a tail estimate of μ. As applications, we estimate convergence rates of approximate solutions of stochastic differential euqations and obtain conditions on the speed of convergence of regularization parameters which guarantee convergence in distribution for the solutions of a random integral equation of the first kind.
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This research was supported in part by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (Project S 32/03).
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Engl, H.W., Wakolbinger, A. Continuity properties of the extension of a locally Lipschitz continuous map to the space of probability measures. Monatshefte für Mathematik 100, 85–103 (1985). https://doi.org/10.1007/BF01295665
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DOI: https://doi.org/10.1007/BF01295665