On tightness of probability measures on Skorokhod spaces

Abstract:

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.