# American Institute of Mathematical Sciences

March  2019, 9(1): 175-190. doi: 10.3934/mcrf.2019010

## Extension of the strong law of large numbers for capacities

 1 School of Mathematics, Shandong University, Jinan 250100, China 2 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China

* Corresponding author: Panyu Wu

Received  March 2016 Revised  January 2018 Published  November 2018

Fund Project: This research is supported by Taishan Scholars Project and the National Natural Science Foundation of China (Grants 11231005, 11601280, 11471190, 11701331 and 11871050), the Natural Science Foundation of Shandong Province of China (Grants ZR2016AQ11 and ZR2016AQ13)

In this paper, with a new notion of exponential independence for random variables under an upper expectation, we establish a kind of strong laws of large numbers for capacities. Our limit theorems show that the cluster points of empirical averages not only lie in the interval between the upper expectation and the lower expectation with lower probability one, but such an interval also is the unique smallest interval of all intervals in which the limit points of empirical averages lie with lower probability one. Furthermore, we also show that the cluster points of empirical averages could reach the upper expectation and lower expectation with upper probability one.

Citation: Zengjing Chen, Weihuan Huang, Panyu Wu. Extension of the strong law of large numbers for capacities. Mathematical Control & Related Fields, 2019, 9 (1) : 175-190. doi: 10.3934/mcrf.2019010
##### References:
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Gilboa, Expected utility theory with purely subjective non-additive probabilities, Journal of Mathematical Economics, 16 (1987), 65-88.  doi: 10.1016/0304-4068(87)90022-X.  Google Scholar [16] P. J. Huber, The use of Choquet capacities in statistics, Bulletin of the International Statistical Institute, 45 (1973), 181-191.   Google Scholar [17] F. Maccheroni and M. Marinacci, A strong law of large number for capacities, The Annals of Probability, 33 (2005), 1171-1178.  doi: 10.1214/009117904000001062.  Google Scholar [18] M. Marinacci, Limit laws for non-additive probabilities and their frequentist interpretation, Journal of Economic Theory, 84 (1999), 145-195.  doi: 10.1006/jeth.1998.2479.  Google Scholar [19] S. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A-Mathematics, 52 (2009), 1391-1411.  doi: 10.1007/s11425-009-0121-8.  Google Scholar [20] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint, arXiv: 1002.4546. Google Scholar [21] Y. Rébillé, Law of large numbers for non-additive measures, Journal of Mathematical Analysis and Applications, 352 (2009), 872-879.  doi: 10.1016/j.jmaa.2008.11.060.  Google Scholar [22] A. N. Shiryaev, Probability, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar [23] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587.  doi: 10.2307/1911053.  Google Scholar [24] P. Terán, Law of large numbers for the possibilistic mean value, Fuzzy Sets and Systems, 245 (2014), 116-124.  doi: 10.1016/j.fss.2013.10.011.  Google Scholar [25] P. Terán, Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.  doi: 10.1090/S0002-9947-2014-06053-4.  Google Scholar [26] P. Terán, Counterexamples to a central limit theorem and a weak law of large numbers for capacities, Statistic and Probability Letters, 96 (2015), 185-189.  doi: 10.1016/j.spl.2014.08.007.  Google Scholar [27] P. P. Wakker, Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle, Econometrica, 69 (2001), 1039-1059.  doi: 10.1111/1468-0262.00229.  Google Scholar [28] P. Walley and T. Fine, Towards a frequentist theory of upper and lower probability, The Annals of Statistics, 10 (1982), 741-761.  doi: 10.1214/aos/1176345868.  Google Scholar [29] L. Wasserman and J. Kadane, Bayes' Theorem for Choquet capacities, The Annals of Statistics, 18 (1990), 1328-1339.  doi: 10.1214/aos/1176347752.  Google Scholar [30] L. Zhang, Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science China Mathematics, 59 (2016), 751-768.  doi: 10.1007/s11425-015-5105-2.  Google Scholar [31] L. Zhang, Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations, preprint, arXiv: 1608.00710. Google Scholar

show all references

##### References:
 [1] H. Agahi, A. Mohammadpour, R. Mesiar and Y. Ouyang, On a strong law of large numbers for momtone measures, Statistics and Probability Letters, 83 (2013), 1213-1218.  doi: 10.1016/j.spl.2013.01.021.  Google Scholar [2] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 3rd edition, Springer, 2006.  Google Scholar [3] R. Badard, The law of large numbers for fuzzy processes and the estimation problem, Information Sciences, 28 (1982), 161-178.  doi: 10.1016/0020-0255(82)90046-9.  Google Scholar [4] X. Chen and Z. Chen, Weak and strong limit theorems for stochastic processes under nonadditive probability, Abstract and Applied Analysis, 2014 (2014), Article ID 645947, 7 pages. doi: 10.1155/2014/645947.  Google Scholar [5] Z. Chen, Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 59 (2016), 945-954.  doi: 10.1007/s11425-015-5095-0.  Google Scholar [6] Z. Chen and L. Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443.  doi: 10.1111/1468-0262.00337.  Google Scholar [7] Z. Chen and R. Kulperger, Minimax pricing and Choquet pricing, Insurance: Mathematics and Economics, 38 (2006), 518-528.  doi: 10.1016/j.insmatheco.2005.11.010.  Google Scholar [8] Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54 (2013), 365-377.  doi: 10.1016/j.ijar.2012.06.002.  Google Scholar [9] F. G. Cozman, Concentration inequalities and laws of large numbers under epistemic and regular irrelevance, International Journal of Approximate Reasoning, 51 (2010), 1069-1084.  doi: 10.1016/j.ijar.2010.08.009.  Google Scholar [10] G. De Cooman and F. Hermans, Imprecise probability trees: Bridging two theories of imprecise probability, Artificial Intelligence, 172 (2008), 1400-1427.  doi: 10.1016/j.artint.2008.03.001.  Google Scholar [11] G. De Cooman and E. Miranda, Weak and strong laws of large numbers for coherent lower previsions, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432.  doi: 10.1016/j.jspi.2007.10.020.  Google Scholar [12] L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Analysis, 34 (2011), 139-161.  doi: 10.1007/s11118-010-9185-x.  Google Scholar [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar [14] L. G. Epstein and D. Schneider, IID: independently and indistinguishably distributed, Journal of Economic Theory, 113 (2003), 32-50.  doi: 10.1016/S0022-0531(03)00121-2.  Google Scholar [15] I. Gilboa, Expected utility theory with purely subjective non-additive probabilities, Journal of Mathematical Economics, 16 (1987), 65-88.  doi: 10.1016/0304-4068(87)90022-X.  Google Scholar [16] P. J. Huber, The use of Choquet capacities in statistics, Bulletin of the International Statistical Institute, 45 (1973), 181-191.   Google Scholar [17] F. Maccheroni and M. Marinacci, A strong law of large number for capacities, The Annals of Probability, 33 (2005), 1171-1178.  doi: 10.1214/009117904000001062.  Google Scholar [18] M. Marinacci, Limit laws for non-additive probabilities and their frequentist interpretation, Journal of Economic Theory, 84 (1999), 145-195.  doi: 10.1006/jeth.1998.2479.  Google Scholar [19] S. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A-Mathematics, 52 (2009), 1391-1411.  doi: 10.1007/s11425-009-0121-8.  Google Scholar [20] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint, arXiv: 1002.4546. Google Scholar [21] Y. Rébillé, Law of large numbers for non-additive measures, Journal of Mathematical Analysis and Applications, 352 (2009), 872-879.  doi: 10.1016/j.jmaa.2008.11.060.  Google Scholar [22] A. N. Shiryaev, Probability, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar [23] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587.  doi: 10.2307/1911053.  Google Scholar [24] P. Terán, Law of large numbers for the possibilistic mean value, Fuzzy Sets and Systems, 245 (2014), 116-124.  doi: 10.1016/j.fss.2013.10.011.  Google Scholar [25] P. Terán, Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.  doi: 10.1090/S0002-9947-2014-06053-4.  Google Scholar [26] P. Terán, Counterexamples to a central limit theorem and a weak law of large numbers for capacities, Statistic and Probability Letters, 96 (2015), 185-189.  doi: 10.1016/j.spl.2014.08.007.  Google Scholar [27] P. P. Wakker, Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle, Econometrica, 69 (2001), 1039-1059.  doi: 10.1111/1468-0262.00229.  Google Scholar [28] P. Walley and T. Fine, Towards a frequentist theory of upper and lower probability, The Annals of Statistics, 10 (1982), 741-761.  doi: 10.1214/aos/1176345868.  Google Scholar [29] L. Wasserman and J. Kadane, Bayes' Theorem for Choquet capacities, The Annals of Statistics, 18 (1990), 1328-1339.  doi: 10.1214/aos/1176347752.  Google Scholar [30] L. Zhang, Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science China Mathematics, 59 (2016), 751-768.  doi: 10.1007/s11425-015-5105-2.  Google Scholar [31] L. Zhang, Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations, preprint, arXiv: 1608.00710. Google Scholar
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