## Abstract

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_{i}i≥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.

Original language | English |
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Pages (from-to) | 445-457 |

Number of pages | 13 |

Journal | Statistics |

Volume | 52 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 4 2018 |

Externally published | Yes |

## Keywords

- Beta process
- nonparametric Bayesian statistics
- random measures
- vague convergence

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty