Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model

Chatterjee, Sourav ; Shao, Qi-Man

Ann. Appl. Probab., Tome 21 (2011) no. 1, p. 464-483 / Harvested from Project Euclid

Résumé

Let (W, W') be an exchangeable pair. Assume that E(W − W'|W) = g(W) + r(W), ¶ where g(W) is a dominated term and r(W) is negligible. Let G(t) = ∫₀^tg(s) ds and define p(t) = c₁e^−c₀G(t), where c₀ is a properly chosen constant and c₁ = 1 / ∫_−∞^∞e^−c₀G(t) dt. Let Y be a random variable with the probability density function p. It is proved that W converges to Y in distribution when the conditional second moment of (W − W') given W satisfies a law of large numbers. A Berry–Esseen type bound is also given. We use this technique to obtain a Berry–Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie–Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli–Laplace Markov chain is also discussed.

Détail

Publié le : 2011-04-15
Classification:  Stein’s method,  exchangeable pair,  Berry–Esseen bound,  Curie–Weiss model,  60F05,  60G09