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.
Categories: math.ST stat.TH