Stein’s method

These lecture notes develop Stein’s method for distributional approximation: a way to compare probability laws and obtain explicit error bounds. They grew out of a course for students in mathematics and statistics at the Université libre de Bruxelles.

The approach starts from identities that characterise a target distribution. Stein equations turn differences of expectations into expressions that can be bounded using the structure of the random variables under study. The notes cover probability metrics, Poisson approximation through the Chen–Stein method, normal approximation, and the interpretation of Stein operators as Markov-process generators.

Lecture notes

Read the full notes (PDF) · Download

Stein’s method, Yvik Swan. Version dated 29 September 2026 · 94 pages · 6.2 MB.

Chapters

1. Introduction to distributional approximation
   Bernoulli sums, Monte Carlo comparisons, error bounds, and probability metrics.
2. Poisson approximation and the Chen–Stein method
   Chen’s identity, Stein operators and equations, approximation bounds, and the generator interpretation.
3. Normal approximation and the Stein method (PDF)
   Gaussian Stein equations, Wasserstein and Kolmogorov bounds, and the generator interpretation.

The first two chapters have concise web versions with interactive figures. The PDF contains the full development, proofs, exercises, and bibliography.

Stein equations and derivative bounds

Solving and bounding Stein equations: a brief account of the kernel method of Rapin and Swan, with the arXiv article and eleven companion PDFs.

Prerequisites

The notes assume courses in measure theory and advanced probability. Some knowledge of functional analysis is helpful.