Abstract
Classical Kolmogorov’s and Rosenthal’s inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers. In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng (2008), we introduce the concept of negative dependence of random variables and establish Kolmogorov’s and Rosenthal’s inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations. As an application, we show that Kolmogorov’s strong law of larger numbers holds for independent and identically distributed random variables under a continuous sub-linear expectation if and only if the corresponding Choquet integral is finite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen Z J. Strong laws of large numbers for capacities. ArXiv:1006.0749, 2010
Denis L, Martini C. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann Appl Probab, 2006, 16: 827–852
Gao F Q, Xu M Z. Large deviations and moderate deviations for independent random variables under sublinear expectations (in Chinese). Sci Sin Math, 2011, 41: 337–352
Gao F Q, Xu M Z. Relative entropy and large deviations under sublinear expectations. Acta Math Sci Ser B, 2012, 32: 1826–1834
Gilboa I. Expected utility theory with purely subjective non-additive probabilities. J Math Econom, 1987, 16: 65–68
Huber P, Strassen V. Minimax tests and the Neyman-Pearson lemma for capacity. Ann Statist, 1973, 1: 252–263
Marinacci M. Limit laws for non-additive probabilities and their frequentist interpretation. J Econom Theory, 1999, 84: 145–195
Matula P. A note on the almost sure convergence of sums of negatively dependent random variables. Statist Probab Lett, 1992, 15: 209–213
Newman C M, Wright A L. An invariance principle for certain dependent sequence. Ann Probab, 1981, 9: 671–675
Peng S. BSDE and related g-expectation. Pitman Res Notes Math Ser, 1997, 364: 141–159
Peng S. Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type. Probab Theory Related Fields, 1999, 113: 473–499
Peng S. G-expectation, G-Brownian motion and related stochastic calculus of Ito type. In: Proceedings of the 2005 Abel Symposium. Berlin-Heidelberg: Springer, 2008, 541–567
Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl, 2008, 118: 2223–2253
Peng S. A new central limit theorem under sublinear expectations. ArXiv:0803.2656v1, 2008
Peng S. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci China Ser A, 2009, 52: 1391–1411
Peng S. Nonlinear expectations and stochastic calculus under uncertainty. ArXiv:1002.4546, 2010
Shao Q M. A Comparison theorem on moment inequalities between negatively associated and independent random variables. J Theort Probab, 2000, 13: 343–356
Su C, Zhao L C, Wang Y B. Moment inequalities and weak convergence for negatively associated sequences. Sci China Ser A, 1997, 40: 172–182
Yuan D M, An J. Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications. Sci China Ser A, 2009, 52: 1887–1904
Zhang L X. A functional central limit theorem for asymptotically negatively dependent random fields. Acta Math Hungar, 2000, 83: 237–259
Zhang L X. A Strassen’s law of the iterated logarithm for negatively associated random vectors. Stoch Process Appl, 2001, 95: 311–328
Zhang L X. The weak convergence for functions of negatively associated random variables. J Multiv Anal, 2001, 78: 272–298
Zhang L X. Donsker’s invariance principle under the sub-linear expectation with an application to Chung’s law of the iterated logarithm. Commun Math Stat, 2015, 3: 187–214
Zhang L X, Wen J W. A weak convergence for negatively associated fields. Statist Probab Lett, 2001, 53: 259–267
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, L. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci. China Math. 59, 751–768 (2016). https://doi.org/10.1007/s11425-015-5105-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-015-5105-2