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On tightness of probability measures on Skorokhod spaces. (English) Zbl 1335.60005

Summary: The equivalences to and the connections between the modulus-of-continuity condition, compact containment and tightness on \( D_{E}[a,b]\) with \( a<b\) are studied. The results within are tools for establishing tightness for probability measures on \( D_E[a,b]\) that generalize and simplify prevailing results in the cases that \( E\) is a metric space, nuclear space dual or, more generally, a completely regular topological space. Applications include establishing weak convergence to martingale problems, the long-time typical behavior of nonlinear filters and particle approximation of cadlag probability-measure-valued processes. This particle approximation is studied herein, where the distribution of the particles is the underlying measure-valued process at an arbitrarily fine discrete mesh of points.

MSC:

60B05 Probability measures on topological spaces
60B10 Convergence of probability measures
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