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

We propose a way of finding a Stein type characterization of a given absolutely continuous distribution $μ$ 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 $μ$. 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.

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