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

Abstract

ABSTRACTThis paper deals with studying vague convergence of random measures of the form , where is a sequence of independent and identically distributed random variables with common distribution Π, 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 μ 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.