On metrizing vague convergence of random measures with applications on Bayesian nonparametric models. Issue 2 (4th March 2018) - Al-Labadi, Luai
ABSTRACT: This paper deals with studying vague convergence of random measures of the formμ n = ∑ i = 1 n p i, n δ θ i, where( θ i ) 1 ≤ i ≤ n is a sequence of independent and identically distributed random variables with common distribution Π, δ θ i denotes the Dirac measure atθ i and( p i, n ) 1 ≤ i ≤ n are random variables, independent of( θ i ) i ≥ 1, chosen according to certain procedures such thatp i, n → p i almost surely, asn → ∞, for fixed i . We show that, asn → ∞, μ n converges vaguely almost surely toμ = ∑ i = 1 ∞ p i δ θ i if and only ifμ n ( k ) = ∑ i = 1 k p i, n δ θ i converges vaguely almost surely toμ ( k ) = ∑ i = 1 k p i δ θ 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.