Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On Metrizing Vague Convergence of Random Measures with Applications on Bayesian Nonparametric Models (1610.03083v1)

Published 10 Oct 2016 in math.ST and stat.TH

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.

Summary

We haven't generated a summary for this paper yet.