This paper deals with studying vague convergence of random measures of the form μn = ∑n i=1 pi,nδΘi, where (Θi)1≤i≤n is a sequence of independent and identically distributed random variables with common distribution Π, δΘi denotes the Dirac measure at Θi and pi,n are random variables, independent of Θii≥1, chosen according to certain procedures such that pi,n → pi almost surely, as n→∞, for fixed i. We show that, as n→∞, μn converges vaguely almost surely to μn = ∑∞ i=1 pi,nδΘi if and only if μn (k)=μn = ∑k i=1 pi,nδΘi converges vaguely almost surely to μ(k)=μn = ∑k i=1 pi,nδΘi 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.
- Beta process
- nonparametric Bayesian statistics
- random measures
- vague convergence
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty