Stein’s method for concentration inequalities

We introduce a version of Stein’s method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical statistics are provided.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  W. Hoeffding.  A Combinatorial Central Limit Theorem , 1951 .

[3]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[4]  C. Stein.  A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[5]  D. Burkholder.  Distribution Function Inequalities for Martingales , 1973 .

[6]  J. Besag.  Statistical Analysis of Non-Lattice Data , 1975 .

[7]  Louis H. Y. Chen.  Poisson Approximation for Dependent Trials , 1975 .

[8]  R. Graham,et al.  Spearman's Footrule as a Measure of Disarray , 1977 .

[9]  E. Bolthausen.  An estimate of the remainder in a combinatorial central limit theorem , 1984 .

[10]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[11]  C. Stein.  Approximate computation of expectations , 1986 .

[12]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[13]  D. K. Pickard.  Inference for Discrete Markov Fields: The Simplest Nontrivial Case , 1987 .

[14]  L. Gordon,et al.  Two moments su ce for Poisson approx-imations: the Chen-Stein method , 1989 .

[15]  Pierre Baldi,et al.  On Normal Approximations of Distributions in Terms of Dependency Graphs , 1989 .

[16]  L. Gordon,et al.  Poisson Approximation and the Chen-Stein Method , 1990 .

[17]  A. Barbour.  Stein's method for diffusion approximations , 1990 .

[18]  A. Barbour,et al.  Poisson Approximation , 1992 .

[19]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[20]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..

[21]  F. Götze,et al.  The Rate of Convergence for Multivariate Sampling Statistics , 1993 .

[22]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[23]  Y. Rinott,et al.  On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics , 1997 .

[24]  G. Reinert,et al.  Stein's method and the zero bias transformation with application to simple random sampling , 1997, math/0510619.

[25]  Jason E. Fulman.  Stein’s method and non-reversible Markov chains , 1997, math/9712241.

[26]  M. Schmuckenschläger.  Curvature of Nonlocal Markov Generators , 1998 .

[27]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[28]  Colin McDiarmid.  Concentration For Independent Permutations , 2002, Comb. Probab. Comput..

[29]  S. Boucheron,et al.  Concentration inequalities using the entropy method , 2003 .

[30]  Colin McDiarmid,et al.  Concentration for locally acting permutations , 2003, Discret. Math..

[31]  Jason E. Fulman.  Stein’s method and Plancherel measure of the symmetric group , 2003, math/0305423.

[32]  Susan Holmes,et al.  Stein's Method: Expository Lectures and Applications , 2004 .

[33]  Van H. Vu,et al.  Divide and conquer martingales and the number of triangles in a random graph , 2004, Random Struct. Algorithms.

[34]  Sourav Chatterjee,et al.  Concentration of Haar measures, with an application to random matrices , 2005 .

[35]  Exchangeable pairs and Poisson approximation , 2004, math/0411525.

[36]  Q. Shao,et al.  The Berry-Esseen bound for character ratios , 2005 .

[37]  P. Collet,et al.  Concentration inequalities for random fields via coupling , 2005, math/0503483.

[38]  S. Chatterjee.  Concentration Inequalities With Exchangeable Pairs , 2005 .

[39]  Martin Raič.  CLT-related large deviation bounds based on Stein's method , 2007, Advances in Applied Probability.