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 μ_n = ∑ⁿ _i=1 pi,nδΘ_i, where (Θ_i)1≤i≤n is a sequence of independent and identically distributed random variables with common distribution Π, δΘ_i denotes the Dirac measure at Θ_i and p_i,n are random variables, independent of Θ_ii≥1, chosen according to certain procedures such that p_i,n → p_i almost surely, as n→∞, for fixed i. We show that, as n→∞, μ_n converges vaguely almost surely to μ_n = ∑^∞ _i=1 pi,nδΘ_i if and only if μ_n ^(k)=μ_n = ∑^k _i=1 pi,nδΘ_i converges vaguely almost surely to μ^(k)=μ_n = ∑^k _i=1 pi,nδΘ_i 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.