Stein's method of exchangeable pairs for the Beta distribution and generalizations

Understand

We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous distribution.

• This distribution is characterized by a first order linear differential Stein operator, whose coefficients $\gamma$ and $\eta$ are motivated by two regression properties satisfied by the pair $(W,W')$.
• Furthermore, the general theory of Stein's method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given.
• This abstract approach is a certain extension of the theory developed in the papers \cite{ChSh} and \cite{EiLo10}, which only consider the framework of the density approach, i.e.