Stein's Method of Exchangeable Pairs with Application to the Curie-Weiss Model

By Sourav Chatterjee, Qi-Man Shao. 2009

135 citations recorded by the source corpus.

Let (W,W ′) be an exchangeable pair. Assume that E(W − W ′|W ) = g(W ) + r(W ), where g(W ) is a dominated term while r(W ) is negligible. Let G(t) = ∫ t 0 g(s)ds and define p(t) = c1e −c0G(t), where c0 is a properly chosen constant and c1 = 1/ ∫∞ −∞ p(t)dt . Let Y be a random variable with the probability density function p. In this talk we shall proved that W converges to Y in distribution under certain regular conditions. A Berry-Esseen type bound is also given. An application to the Curie-Weiss model will be discussed. The talk is based on a joint work with Sourav Chatterjee.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/95ebf16282c5de948c78a927b5942ad87eec6119

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Citations in the corpus

Stein's method for dependent random variables occurring in Statistical Mechanics (2009, methodology): “Shao about Stein’s method with applications to the Curie-Weiss model [4].”
Stein's method for comparison of univariate distributions (2014, methodology): “Of course one could proceed as in [28, 21] or [65] by imposing ad hoc assumptions on the target density which ensure that the functions in (62) have good properties.”
Fundamentals of Stein's method (2011, methodology): “An alternative approach (followed in [21, 22]) is to use exchangeable pairs with a modified a-Stein condition.”
Applications of Stein's method for concentration inequalities (2009, result): “This limit theorem was first proved by Simon and Griffiths [35] and error bounds were obtained recently [13, 16].”
Measuring Sample Quality with Stein's Method (2015, background): “P ,Gk·k) converges to zero only if the Wasserstein distance dWk·k(μm, P ) does [10, 11].”
A short survey of Stein's method (2014, methodology): “Stein’s method for a mixture of two normal distributions, with an application to spin glasses, appeared in my 2010 paper [23], while another non-normal distribution arising at the critical temperature of the Curie-Weiss model of ferromagnets was tackled in my joint paper with Shao [30] in 2011 and i”
Approximation of stable law in Wasserstein-1 distance by Stein’s method (2017, methodology): “Nowadays, Stein’s method has been extended and refined by many authors and become a very important tool for getting bounds of measure approximations, see [6, 10, 14, 22, 24, 29, 39, 46, 26, 27, 28, 33, 42, 43, 44, 47].”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, background): “, [24]), which could be of independent interest.”
Measuring Sample Quality with Diffusions (2016, background): “, [12, 11] for explicit bounds or [53] for a recent review).”