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Central moment inequalities using Stein’s method. (English) Zbl 1459.60045
Summary: We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size-bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare them to those existing in related settings. Finally, we illustrate the power of the theory by bounding the central moments of sums of neighbourhood statistics in sparse Erdős-Rényi random graphs.
60E15 Inequalities; stochastic orderings
60C05 Combinatorial probability
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[1] R. Arratia and P. Baxendale, 2015. Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior. Probab. Theory Related Fields, 162(3-4):411-429. · Zbl 1323.60034
[2] A. D. Barbour, L. Holst, and S. Janson, 1992. Poisson approximation, volume 2 of Oxford Studies in Probability. Oxford University Press, Oxford. · Zbl 0746.60002
[3] A. D. Barbour, A. Röllin, and N. Ross, 2019. Error bounds in local limit theorems using Stein’s method. Bernoulli, 25(2):1076-1104. · Zbl 07049400
[4] N. Batir, 2008. Inequalities for the gamma function. Arch. Math. (Basel), 91(6):554-563. · Zbl 1165.33001
[5] S. Boucheron, G. Lugosi, and P. Massart, 2003. Concentration inequalities using the entropy method. Ann. Probab., 31(3):1583-1614. · Zbl 1051.60020
[6] S. Boucheron, O. Bousquet, G. Lugosi, and P. Massart, 2005. Moment inequalities for functions of independent random variables. Ann. Probab., 33(2):514-560. · Zbl 1074.60018
[7] S. Boucheron, G. Lugosi, and P. Massart, 2013. Concentration inequalities. Oxford University Press, Oxford. A nonasymptotic theory of independence, With a foreword by Michel Ledoux. · Zbl 1279.60005
[8] S. Chatterjee, 2007. Stein’s method for concentration inequalities. Probab. Theory Related Fields, 138(1-2):305-321. · Zbl 1116.60056
[9] S. Chatterjee, 2014. A short survey of Stein’s method. In S. Y. Jang, Y. R. Kim, D.-W. Lee, and I. Yie, editors, Proceedings of the International Congress of Mathematicians, Seoul 2014, Volume IV, Invited Lectures, pages 1-24, Seoul, Korea. KYUNG MOON SA Co. Ltd. · Zbl 1373.60052
[10] S. Chatterjee and P. S. Dey, 2010. Applications of Stein’s method for concentration inequalities. Ann. Probab., 38(6):2443-2485. · Zbl 1203.60023
[11] L. H. Y. Chen and A. Röllin, 2010. Stein couplings for normal approximation. Preprint arXiv:1003.6039v2.
[12] L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, 2011. Normal approximation by Stein’s method. Probability and its Applications (New York). Springer, Heidelberg.
[13] L. H. Y. Chen, X. Fang, and Q.-M. Shao, 2013. From Stein identities to moderate deviations. Ann. Probab., 41(1):262-293. · Zbl 1275.60029
[14] N. Cook, L. Goldstein, and T. Johnson, 2018. Size biased couplings and the spectral gap for random regular graphs. Ann. Probab., 46(1):72-125. · Zbl 1386.05105
[15] S. Ghosh and L. Goldstein, 2011. Concentration of measures via size-biased couplings. Probab. Theory Related Fields, 149(1-2):271-278. · Zbl 1239.60011
[16] S. Ghosh and L. Goldstein, 2011. Applications of size biased couplings for concentration of measures. Electron. Commun. Probab., 16:70-83. · Zbl 1227.60021
[17] S. Ghosh, L. Goldstein, and M. Raic, 2011. Concentration of measure for the number of isolated vertices in the Erdos-Rényi random graph by size bias couplings. Statist. Probab. Lett., 81(11):1565-1570. · Zbl 1226.05227
[18] S. Janson, 2004. Large deviations for sums of partly dependent random variables. Random Structures Algorithms, 24(3):234-248. · Zbl 1044.60021
[19] L. Mackey, M. I. Jordan, R. Y. Chen, B. Farrell, and J. A. Tropp, 2014. Matrix concentration inequalities via the method of exchangeable pairs. Ann. Probab., 42(3):906-945. · Zbl 1294.60008
[20] P. Massart, 2007. Concentration inequalities and model selection, volume 1896 of Lecture Notes in Mathematics. Springer, Berlin. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6-23, 2003, With a foreword by Jean Picard.
[21] C. McDiarmid, 1998. Concentration. In Probabilistic methods for algorithmic discrete mathematics, volume 16 of Algorithms Combin., pages 195-248. Springer, Berlin.
[22] D. Paulin, L. Mackey, and J. A. Tropp, 2016. Efron-Stein inequalities for random matrices. Ann. Probab., 44(5):3431-3473. · Zbl 1378.60025
[23] H. Robbins, 1955. A remark on Stirling’s formula. Amer. Math. Monthly, 62:26-29. · Zbl 0068.05404
[24] A. Röllin, 2007. Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab., 17(5-6):1596-1614. · Zbl 1143.60020
[25] R.
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