On Metrizing Vague Convergence of Random Measures with Applications on Bayesian Nonparametric Models

Abstract

This paper deals with studying vague convergence of random measures of the form $\mu_{n}=\sum_{i=1}^{n} p_{i,n} \delta_{\theta_i}$, where $(\theta_i)_{1\le i \le n}$ is a sequence of independent and identically distributed random variables with common distribution $\Pi$, $(p_{i,n})_{1 \le i \le n}$ are random variables chosen according to certain procedures and are independent of $(\theta_i)_{i \geq 1}$ and $\delta_{\theta_i}$ denotes the Dirac measure at $\theta_i$. We show that $\mu_{n}$ converges vaguely to $\mu=\sum_{i=1}^{\infty} p_{i} \delta_{\theta_i}$ if and only if $\mu^{(k)}_{n}=\sum_{i=1}^{k} p_{i,n} \delta_{\theta_i}$ converges vaguely to $\mu^{(k)}=\sum_{i=1}^{k} p_{i} \delta_{\theta_i}$ for all $k$ fixed. The limiting process $\mu$ 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.