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November 2010 Applications of Stein’s method for concentration inequalities
Sourav Chatterjee, Partha S. Dey
Ann. Probab. 38(6): 2443-2485 (November 2010). DOI: 10.1214/10-AOP542

Abstract

Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Rényi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

Citation

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Sourav Chatterjee. Partha S. Dey. "Applications of Stein’s method for concentration inequalities." Ann. Probab. 38 (6) 2443 - 2485, November 2010. https://doi.org/10.1214/10-AOP542

Information

Published: November 2010
First available in Project Euclid: 24 September 2010

zbMATH: 1203.60023
MathSciNet: MR2683635
Digital Object Identifier: 10.1214/10-AOP542

Subjects:
Primary: 60E15 , 60F10
Secondary: 60C05 , 82B44

Keywords: concentration inequality , Curie–Weiss model , Erdős–Rényi random graph , exponential random graph , Gibbs measures , Ising model , large deviation , Stein’s method

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 6 • November 2010
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