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Mathematica Slovaca

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Volume 65, Issue 6

Issues

Exponential Inequalities for Bounded Random Variables

Guangyue Huang
  • Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information Science Henan Normal University 453007 Henan CHINA
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/ Xin Guo / Hongxia Du
  • Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information Science Henan Normal University 453007 Henan CHINA
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/ Yi He
  • Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information Science Henan Normal University 453007 Henan CHINA
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/ Yu Miao
  • Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information Science Henan Normal University 453007 Henan CHINA
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Published Online: 2016-02-09 | DOI: https://doi.org/10.1515/ms-2015-0106

Abstract

In the paper, several precise exponential inequalities for the sums of bounded or semi-bounded random variables are established, which involve independent random variables, martingale difference sequence, negatively associated random variables, Markov chains.

Keywords: bounded random variables; martingale difference sequence; negatively associated; Markov chians; exponential inequalities

References

  • [1] ALAM, K.-SAXENA, K. M. L.: Positive dependent in multivariate distributions, Comm. Statist. Theory Methods 10 (1981), 1183-1196.Google Scholar

  • [2] ASMUSSEN, S. et al: Stationarity detection in the initial transient problem, ACMTrans. Modeling Comput. Simulation 2 (1992), 130-157.Google Scholar

  • [3] AZUMA, K.: Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), 357-367.Google Scholar

  • [4] BENTKUS, V.: An inequality for large deviation probabilities of sums of bounded i.i.d. r.v, Lith. Math. J. 41 (2001), 144-153.Google Scholar

  • [5] BENTKUS, V.: An inequality for tail probabilities of martingales with differences bounded from one side, J. Theoret. Probab. 16 (2003), 161-173.CrossrefGoogle Scholar

  • [6] BENTKUS, V.: On measure concentration for separately Lipschitz functions in product spaces, Israel J. Math. 158 (2007), 1-17.Google Scholar

  • [7] BLOCK, H. M. et al: Some concepts of negative dependence, Ann. Probab. 10 (1982), 765-772.CrossrefGoogle Scholar

  • [8] DE LA PEÑA, V. H.: A bound on the moment generating function of a sum of dependent variables with an application to simple random sampling without replacement, Ann. Inst. Henri Poincaré Probab. Stat. 30 (1994), 197-211.Google Scholar

  • [9] DE LA PEÑA, V. H.: A general class of exponential inequalities for martingales and ratios, Ann. Probab. 27 (1999), 537-564.CrossrefGoogle Scholar

  • [10] DJELLOUT, H. et al: Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab. 32 (2004), 2702-2732.CrossrefGoogle Scholar

  • [11] DOOB, J. L.: Stochastic Processes, Wiley, New York, 1953.Google Scholar

  • [12] GLYNN, P. W.-ORMONEIT, D.: Hoeffding’s inequality for uniformly ergodic Markov chains, Statist. Probab. Lett. 56 (2002), 143-146.CrossrefGoogle Scholar

  • [13] HOEFFDING, W.: Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13-30.CrossrefGoogle Scholar

  • [14] JOAG-DEV, K.-PROSCHAN, F.: Negative association of random variables with applications, Ann. Statist. 11 (1983), 286-295.CrossrefGoogle Scholar

  • [15] KIM, T. S.-KIM, H. C.: On the exponential inequality for negatively dependent sequence, Commun. Korean Math. Soc. 22 (2007), 315-321.Google Scholar

  • [16] MATUłA, P.: A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett. 15 (1992), 209-213.CrossrefGoogle Scholar

  • [17] MEYN, S. P.-TWEEDIE R. L.: Markov Chains and Stochastic Stability, Springer, New York, 1993.Google Scholar

  • [18] MIAO, Y.: A note on the martingale inequality, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), Art. 187.Google Scholar

  • [19] MIAO, Y.: Concentration inequalities for semi-bounded martingales, ESAIM Probab. Statist. 12 (2008), 51-57.CrossrefGoogle Scholar

  • [20] MIAO, Y. et al: Deviation inequalities for the estimator of linear parameter in stochastic processes, Comm. Statist. Theory Methods 36 (2007), 2263-2272.Google Scholar

  • [21] HOOGHABI, H. J.-AZARNOOSH H. A.: Exponential inequality for negatively associated random variables, Statist. Papers 50 (2009), 419-428.CrossrefGoogle Scholar

  • [22] PINELIS, I.: Extremal probabilistic problems and Hotelling’s T2 test under a symmetry assumption, Ann. Statist. 22 (1994), 357-368.CrossrefGoogle Scholar

  • [23] PINELIS, I.: Optimal tail comparison based on comparison of moments. In: High Dimensional Probability (Oberwolfach, 1996). Progr. Probab. 43, Birkhäuser, Basel, 1998, pp. 297-314.Google Scholar

  • [24] PINELIS, I.: Exact inequalities for sums of asymmetric random variables, with applications, Probab. Theory Related Fields 139 (2007), 605-635.Google Scholar

  • [25] PINELIS, I.: On inequalities for sums of bounded random variables, J. Math. Inequal. 2 (2008), 1-7.CrossrefGoogle Scholar

  • [26] PUTERMAN, M. L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, New York, 1994.Google Scholar

  • [27] ROUSSAS, G. G.: Exponential probability inequalities with some applications, IMS Lecture Notes Monogr. Ser. 30 (1996), 303-319.Google Scholar

  • [28] SHAO, Q. M.: A comparison theorem on maximal inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), 343-356.CrossrefGoogle Scholar

  • [29] TALAGRAND, M.: The missing factor in Hoeffding’s inequalities, Ann. Inst. Henri Poincaré Probab. Stat. 31 (1995), 689-702.Google Scholar

  • [30] STEIGER, W. L.: Some Kolmogoroff-type inequalities for bounded random variables, Biometrika 54 (1967), 641-647. CrossrefGoogle Scholar

About the article

Received: 2012-05-06

Accepted: 2013-02-13

Published Online: 2016-02-09

Published in Print: 2015-12-01


Citation Information: Mathematica Slovaca, Volume 65, Issue 6, Pages 1557–1570, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0106.

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