A short survey of Stein's method

Abstract

Stein's method is a powerful technique for proving central limit theorems in probability theory when more straightforward approaches cannot be implemented easily. This article begins with a survey of the historical development of Stein's method and some recent advances. This is followed by a description of a "general purpose" variant of Stein's method that may be called the generalized perturbative approach, and an application of this method to minimal spanning trees. The article concludes with the descriptions of some well known open problems that may possibly be solved by the perturbative approach or some other variant of Stein's method.

arXiv commentContribution to the Proceedings of the ICM 2014. 25 pages

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• Stein operators, kernels and discrepancies for multivariate continuous distributions
• Second order Stein: SURE for SURE and other applications in high-dimensional inference
• A diffusion approach to Stein's method on Riemannian manifolds
• High order steady-state diffusion approximation of the Erlang-C system
• Approximation of Riemannian measures by Stein's method
• Stein's method for dynamical systems
• Central moment inequalities using Stein's method
• A quantitative central limit theorem for the effective conductance on the discrete torus
• Constructing exchangeable pairs by diffusion on manifolds and its application
• Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels
• Concentration inequalities from monotone couplings for graphs, walks, trees and branching processes
• Causal Inference in Longitudinal Data under Unknown Interference
• Dirichlet approximation of equilibrium distributions in Cannings models with mutation
• Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods
• Non-uniform Berry-Esseen Bound by Unbounded Exchangeable Pair Approach