Stein's Method

Stein's method is a technique which allows to transform the problem of bounding the error term in stochastic approximations into the problem of bounding suitably chosen differential operators. Initially proposed for Gaussian approximation, there now exist applications to many kinds of stochastic approximation problems ranging from the classical (Poisson, gamma, binomial) to the exotic (Kummer-U, variance Gamma).

The already quite large literature on Stein's method is growing at a steady pace. There exist numerous references for Poisson and Gaussian approximations and we refer to the relevant books and surveys for an overview of this literature. There also exists an important area of application around the fourth moment theorem (the so-called Nourdin-Peccati analysis); we refer to [NP12] or to Ivan Nourdin's remarkable Malliavin-Stein page for a comprehensive overview of this aspect of the literature.

The foundation of Stein's method for any application to any target distribution relies on three fundamental steps :

(A) the identification of suitable differential (or difference) operators acting on sufficiently large classes of functions,

(B) solutions of the corresponding differential equations, and

(C) bounds on these solutions.

Operators mentioned in (A) are known as Stein operators; the differential equations mentioned in (B) are known as Stein equations and the corresponding bounds mentioned in (C) are sometimes called Stein (magic) factors.

Why this page?

Two reasons.

Reason 1 : The three points (A), (B) and (C) described above have now been provided for a large number of different distributions. There is, to the best of my knowledge, no single resource pointing towards all these different results. The aim of the page "Distributions" is to provide such a starting point for whomever seeks to have an overview of this important literature. For obvious reasons we choose to leave out the Gaussian and Poisson frameworks. I aim to list all other targets.

Reason 2 : The tools appearing in Stein's method are now popping up in a quite surprising number of areas, from information theory to sparse estimation. The page "Applications" aims at giving pointers to all the different fields of research wherein Stein's ideas have begun playing an important role.

About the author :

I am Yvik Swan, junior prof at Liège University (Liège is in Belgium, the country of great music, good chocolates and even better beers).

Disclaimer : I am sure that many references are missing/incorrect/badly-labeled. Please do not hesitate to send me a mail to signal anything wrong. You are also invited to use the "comments" section below (you need a google account for this).