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A Sharp Uniform Bound for the Distribution of Sums of Bernoulli Trials

Part of: Distribution theory - Probability Combinatorial probability

Published online by Cambridge University Press:  07 July 2015

JEAN-BERNARD BAILLON ,
ROBERTO COMINETTI  and
JOSÉ VAISMAN
JEAN-BERNARD BAILLON
Affiliation:
SAMM–EA 4543, Université de Paris 1, 75013 Paris, France (e-mail: baillon@univ.paris1.fr)
ROBERTO COMINETTI
Affiliation:
Departamento de Ingeniería Industrial, Universidad de Chile, Avenida República 701, Santiago, Chile (e-mail: cominetti.roberto@gmail.com)
JOSÉ VAISMAN
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, Avenida Blanco Encalada 2120, Santiago, Chile (e-mail: hellovaisman@gmail.com)
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Abstract

In this note we establish a uniform bound for the distribution of a sum Sn=X1+···+Xn of independent non-homogeneous Bernoulli trials. Specifically, we prove that σn(Sn = j) ≤ η, where σn denotes the standard deviation of Sn, and η is a universal constant. We compute the best possible constant η ~ 0.4688 and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for n and j fixed. An application to estimate the rate of convergence of Mann's fixed-point iterations is presented.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60E05: Distributions 60E15: Inequalities; stochastic orderings
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 352 - 361
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Alon, N. and Spencer, J. (1992) The Probabilistic Method, Wiley.Google Scholar
[2] Azuma, K. (1967) Weighted sums of certain dependent random variables. Tohoku Math. J. 19 357–367.CrossRefGoogle Scholar
[3] Baillon, J. B. and Bruck, R. (1996) The rate of asymptotic regularity is $O(1/\sqrt{n})$ . In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (Karsatos, A. G., ed.), Vol. 178 of Lecture Notes in Pure and Applied Mathematics, CRC Press, pp. 51–81.Google Scholar
[4] Baillon, J-B., Cominetti, R. and Vaisman, J. (2013) A sharp uniform bound for the distribution of sums of Bernoulli trials. arXiv:0806.2350v4Google Scholar
[5] Bernstein, S. N. (1937) On certain modifications of Chebyshev's inequality. Doklady Akad. Nauk SSSR 17 275–277.Google Scholar
[6] Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493–507.CrossRefGoogle Scholar
[7] Cominetti, R., Correa, J., Rothvoß, T. and San Martín, J. (2010) Optimal selection of customers for a last-minute offer. Oper. Res. 58 878–888.Google Scholar
[8] Cominetti, R., Soto, J. and Vaisman, J. (2014) On the rate of convergence of Krasnosel'skiǐ–Mann iterations and their connection with sums of Bernoullis. Israel J. Math. 199 757–772.Google Scholar
[9] Darroch, J. N. (1964) On the distribution of the number of successes in independent trials. Ann. Math. Statist. 35 1317–1321.CrossRefGoogle Scholar
[10] Davis, B. and McDonald, D. (1995) An elementary proof of the local central limit theorem. J. Theoret. Probab. 8 693–701.Google Scholar
[11] Feller, W. (1943) Generalization of a probability limit theorem of Cramér. Trans. Amer. Math. Soc. 54 361–372.Google Scholar
[12] Figiel, T., Hitczenko, P., Johnson, W. B., Schechtman, G. and Zinn, J. (1997) Extremal properties of Rademacher functions with applications to the Kintchine and Rosenthal inequalities. Trans. Amer. Math. Soc. 349 997–1027.Google Scholar
[13] Gamkrelidze, N. G. (1988) Application of a smoothness function in the proof of a local limit theorem (in Russian). Teor. Veroyatnost. i Primenen. 33 373–376. Translation in Theory Probab. Appl. 33 (1989) 352–355.Google Scholar
[14] Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables, Addison-Wesley.Google Scholar
[15] Godbole, A. P. and Hitczenko, P. (1998) Beyond the method of bounded differences. In Microsurveys in Discrete Probability (Aldous, D. and Propp, J., eds), Vol. 41 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., AMS, pp. 43–58.CrossRefGoogle Scholar
[16] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[17] Ibragimov, R. and Sharakhmetov, S. (1997) On an exact constant for the Rosenthal inequality. Teor. Veroyatnost. i Primenen. 42 341–350.Google Scholar
[18] Jogdeo, K. and Samuels, S. M. (1968) Monotone convergence of binomial probabilities and a generalization of Ramanujan's equation. Ann. Math. Statist. 39 1191–1195.Google Scholar
[19] Kintchine, A. Y. (1937) Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze. Rec. Mat (Mat. Sbornik) N.S. 2 79–120.Google Scholar
[20] Krasnosel'ski, M. A. (1955) Two remarks on the method of successive approximations. Uspekhi Mat. Nauk 10 123–127.Google Scholar
[21] Mann, W. R. (1953) Mean value methods in iteration. Proc. Amer. Math. Soc. 4 506–510.CrossRefGoogle Scholar
[22] McDiarmid, C. (1989) On the method of bounded differences. In Surveys in Combinatorics, 1989 (Siemons, J., ed.), Vol. 141 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 148–188.Google Scholar
[23] McDonald, D. (1980) On local limit theorem for integer-valued random variables. Theory Probab. Appl. 24 613–619.Google Scholar
[24] Petrov, V. (1995) Limit Theorems of Probability Theory, Vol. 4 of Oxford Studies in Probability, Clarendon Press.Google Scholar
[25] Rogozin, B. A. (1961) An estimate for concentration functions. Theory Probab. Appl. 6 94–97.CrossRefGoogle Scholar
[26] Samuels, S. M. (1965) On the number of successes in independent trials. Ann. Math. Statist. 36 1272–1278.Google Scholar
[27] Schechtman, G. (2007) Extremal configurations for moments of sums of independent positive random variables. In Banach Spaces and their Applications in Analysis (Randrianantoanina, B. et al., eds), de Gruyter, pp. 183–191.Google Scholar
[28] Utev, S. A. (1985) Extremal problems in moment inequalities. In Limit Theorems of Probability Theory, Trudy Inst. Mat., Nauka Sibirsk. Otdel. Novosibirsk, pp. 56–75.Google Scholar
[29] Vaisman, J. (2005) Convergencia Fuerte del Método de Medias Sucesivas para Operadores Lineales No-Expansivos, Memoria de Ingeniería Civil Matemática, Universidad de Chile.Google Scholar