Applications of Stein's method for concentration inequalities

Details Applications of Stein's method for concentration inequalities Applications of Stein's method for concentration inequalities

2009-06-05

arxiv.org/abs/0906.1034v3

Annals of Probability 2010, Vol. 38, No. 6, 2443-2485

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP542 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP542

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 Erd\H{o}s--R\'{e}nyi random graph $G(n,p)$ when $p\ge0.31$. 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.

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