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 ] D_{E}[a,b] with a > b a>b are studied. The results within are tools for establishing tightness for probability measures on D E [ a , b ] D_E[a,b] that generalize and simplify prevailing results in the cases that E 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.