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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, doi: 10.3934/mcrf.2019010
References:
[1]

H. AgahiA. MohammadpourR. 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.

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 3rd edition, Springer, 2006.

[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.

[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.

[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.

[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.

[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.

[8]

Z. ChenP. 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.

[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.

[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.

[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.

[12]

L. DenisM. 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.

[13]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.

[14]

L. G. Epstein and D. Schneider, IID: independently and indistinguishably distributed, Journal of Economic Theory, 113 (2003), 32-50.

[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.

[16]

P. J. Huber, The use of Choquet capacities in statistics, Bulletin of the International Statistical Institute, 45 (1973), 181-191.

[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.

[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.

[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.

[20]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint, arXiv:1002.4546.

[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.

[22]

A. N. Shiryaev, Probability, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-1-4757-2539-1.

[23]

D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587. doi: 10.2307/1911053.

[24]

P. Terán, Law of large numbers for the possibilistic mean value, Fuzzy Sets and Systems, 245 (2014), 116-124.

[25]

P. Terán, Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.

[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.

[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.

[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.

[29]

L. Wasserman and J. Kadane, Bayes' Theorem for Choquet capacities, The Annals of Statistics, 18 (1990), 1328-1339. doi: 10.1214/aos/1176347752.

[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.

[31]

L. Zhang, Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations, preprint, arXiv:1608.00710.

show all references

References:
[1]

H. AgahiA. MohammadpourR. 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.

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, 3rd edition, Springer, 2006.

[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.

[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.

[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.

[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.

[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.

[8]

Z. ChenP. 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.

[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.

[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.

[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.

[12]

L. DenisM. 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.

[13]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.

[14]

L. G. Epstein and D. Schneider, IID: independently and indistinguishably distributed, Journal of Economic Theory, 113 (2003), 32-50.

[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.

[16]

P. J. Huber, The use of Choquet capacities in statistics, Bulletin of the International Statistical Institute, 45 (1973), 181-191.

[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.

[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.

[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.

[20]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint, arXiv:1002.4546.

[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.

[22]

A. N. Shiryaev, Probability, 2nd edition, Springer-Verlag, 1996. doi: 10.1007/978-1-4757-2539-1.

[23]

D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587. doi: 10.2307/1911053.

[24]

P. Terán, Law of large numbers for the possibilistic mean value, Fuzzy Sets and Systems, 245 (2014), 116-124.

[25]

P. Terán, Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.

[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.

[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.

[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.

[29]

L. Wasserman and J. Kadane, Bayes' Theorem for Choquet capacities, The Annals of Statistics, 18 (1990), 1328-1339. doi: 10.1214/aos/1176347752.

[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.

[31]

L. Zhang, Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations, preprint, arXiv:1608.00710.

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