Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model

TL;DR

This paper develops a Stein's method approach using exchangeable pairs to characterize and analyze the convergence of distributions, with specific focus on Beta distributions and applications to Polya urn models.

Contribution

It introduces a new Stein's method framework based on exchangeable pairs for absolutely continuous distributions, including Beta distributions, and applies it to Polya urn convergence analysis.

Findings

Established a Stein characterization for absolutely continuous distributions.

Derived convergence rates for Polya urn models.

Extended exchangeable pairs approach to new distribution classes.

Abstract

We propose a way of finding a Stein type characterization of a given absolutely continuous distribution $\mu$ on $\R$ which is motivated by a regression property satisfied by an exchangeable pair $(W,W')$ where $\calL(W)$ is supposed or known to be close to $\mu$. We also develop the exchangeable pairs approach within this setting. This general procedure is then specialized to the class of Beta distributions and as an application, a convergence rate for the relative number of drawn red balls among the first $n$ drawings from a Polya urn is computed.