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.
References
[1] ALAM, K.-SAXENA, K. M. L.: Positive dependent in multivariate distributions, Comm. Statist. Theory Methods 10 (1981), 1183-1196.10.1080/03610928108828102Search in Google Scholar
[2] ASMUSSEN, S. et al: Stationarity detection in the initial transient problem, ACMTrans. Modeling Comput. Simulation 2 (1992), 130-157.10.1145/137926.137932Search in Google Scholar
[3] AZUMA, K.: Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967), 357-367.10.2748/tmj/1178243286Search in 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), 144-153.10.1023/A:1011668015105Search in Google Scholar
[5] BENTKUS, V.: An inequality for tail probabilities of martingales with differences bounded from one side, J. Theoret. Probab. 16 (2003), 161-173.10.1023/A:1022234622381Search in Google Scholar
[6] BENTKUS, V.: On measure concentration for separately Lipschitz functions in product spaces, Israel J. Math. 158 (2007), 1-17.10.1007/s11856-007-0001-2Search in Google Scholar
[7] BLOCK, H. M. et al: Some concepts of negative dependence, Ann. Probab. 10 (1982), 765-772.10.1214/aop/1176993784Search in Google 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.Search in Google Scholar
[9] DE LA PEÑA, V. H.: A general class of exponential inequalities for martingales and ratios, Ann. Probab. 27 (1999), 537-564.Search in Google Scholar
[10] DJELLOUT, H. et al: Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab. 32 (2004), 2702-2732.10.1214/009117904000000531Search in Google Scholar
[11] DOOB, J. L.: Stochastic Processes, Wiley, New York, 1953.Search in Google Scholar
[12] GLYNN, P. W.-ORMONEIT, D.: Hoeffding’s inequality for uniformly ergodic Markov chains, Statist. Probab. Lett. 56 (2002), 143-146.10.1016/S0167-7152(01)00158-4Search in Google Scholar
[13] HOEFFDING, W.: Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13-30.10.1080/01621459.1963.10500830Search in Google Scholar
[14] JOAG-DEV, K.-PROSCHAN, F.: Negative association of random variables with applications, Ann. Statist. 11 (1983), 286-295.10.1214/aos/1176346079Search in Google Scholar
[15] KIM, T. S.-KIM, H. C.: On the exponential inequality for negatively dependent sequence, Commun. Korean Math. Soc. 22 (2007), 315-321.10.4134/CKMS.2007.22.2.315Search in Google Scholar
[16] MATUłA, P.: A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett. 15 (1992), 209-213.10.1016/0167-7152(92)90191-7Search in Google Scholar
[17] MEYN, S. P.-TWEEDIE R. L.: Markov Chains and Stochastic Stability, Springer, New York, 1993.10.1007/978-1-4471-3267-7Search in Google Scholar
[18] MIAO, Y.: A note on the martingale inequality, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), Art. 187.Search in Google Scholar
[19] MIAO, Y.: Concentration inequalities for semi-bounded martingales, ESAIM Probab. Statist. 12 (2008), 51-57.10.1051/ps:2007033Search in Google Scholar
[20] MIAO, Y. et al: Deviation inequalities for the estimator of linear parameter in stochastic processes, Comm. Statist. Theory Methods 36 (2007), 2263-2272.10.1080/03610920701215266Search in Google Scholar
[21] HOOGHABI, H. J.-AZARNOOSH H. A.: Exponential inequality for negatively associated random variables, Statist. Papers 50 (2009), 419-428.10.1007/s00362-007-0081-4Search in Google Scholar
[22] PINELIS, I.: Extremal probabilistic problems and Hotelling’s T2 test under a symmetry assumption, Ann. Statist. 22 (1994), 357-368.10.1214/aos/1176325373Search in Google 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. 297-314.10.1007/978-3-0348-8829-5_19Search in Google Scholar
[24] PINELIS, I.: Exact inequalities for sums of asymmetric random variables, with applications, Probab. Theory Related Fields 139 (2007), 605-635.10.1007/s00440-007-0055-4Search in Google Scholar
[25] PINELIS, I.: On inequalities for sums of bounded random variables, J. Math. Inequal. 2 (2008), 1-7.10.7153/jmi-02-01Search in Google Scholar
[26] PUTERMAN, M. L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, New York, 1994.10.1002/9780470316887Search in Google Scholar
[27] ROUSSAS, G. G.: Exponential probability inequalities with some applications, IMS Lecture Notes Monogr. Ser. 30 (1996), 303-319.10.1214/lnms/1215453579Search in Google Scholar
[28] SHAO, Q. M.: A comparison theorem on maximal inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), 343-356.10.1023/A:1007849609234Search in Google Scholar
[29] TALAGRAND, M.: The missing factor in Hoeffding’s inequalities, Ann. Inst. Henri Poincaré Probab. Stat. 31 (1995), 689-702.Search in Google Scholar
[30] STEIGER, W. L.: Some Kolmogoroff-type inequalities for bounded random variables, Biometrika 54 (1967), 641-647. 10.1093/biomet/54.3-4.641Search in Google Scholar
Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
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- On Primary Ideals in Posets
- Characterization of the Set of Regular Elements in Ordered Semigroups
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