On metrizing vague convergence of random measures with applications on Bayesian nonparametric models

This paper deals with studying vague convergence of random measures of the form is a sequence of independent and identically distributed random variables with common distribution , denotes the Dirac measure at and are random variables, independent of, chosen according to certain procedures such that almost surely, as, for fixed i. We show that, as converges vaguely almost surely to if and only if converges vaguely almost surely to for all k fixed. The limiting process plays a central role in many areas in statistics, including Bayesian nonparametric models. A finite approximation of the beta process is derived from the application of this result. A simulated example is incorporated, in which the proposed approach exhibits an excellent performance over several existing algorithms.