# American Institute of Mathematical Sciences

September  2015, 5(3): 435-452. doi: 10.3934/mcrf.2015.5.435

## Strong law of large numbers for upper set-valued and fuzzy-set valued probability

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

Received  October 2014 Revised  February 2015 Published  July 2015

In this paper, we introduce the concepts of upper-lower set-valued probabilities and related upper-lower expectations for random variables. With a new concept of independence for random variables, we show a strong law of large numbers for upper-lower set-valued probabilities. Furthermore, we extend those concepts and theorem to the case of fuzzy-set.
Citation: Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control and Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435
##### References:
 [1] Z. Artsteun, Set-valued measures, Transactions of the American Mathematical Society, 165 (1972), 103-125. doi: 10.1090/S0002-9947-1972-0293054-4. [2] Z. Chen and L. Epstein, Ambiguity, risk and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337. [3] Z. Chen and P. Wu, Strong laws of large numbers for Bernoulli experiments under ambiguity, Advances in Intelligent and Soft Computing, 100 (2011), 19-30. doi: 10.1007/978-3-642-22833-9_2. [4] Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilies, International Journal of Approximate Reasoning, 54 (2013), 365-377. doi: 10.1016/j.ijar.2012.06.002. [5] G. Cooman and E. Miranda, Weak and strong laws of large numbers for coherent lower precision, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432. doi: 10.1016/j.jspi.2007.10.020. [6] G. Debereu and D. Schmeidler, The Radom-Nikodym derivative of a correspondence, Proc. Sixth Berkeley Sympo. Math. Statist. Probab, Univ. of California Press, 2 (1970), 41-56. [7] L. DeRobertis and J. A. Hartigan, Bayesian inference using intervals of measures, Annals of Statistics, 9 (1981), 235-244. doi: 10.1214/aos/1176345391. [8] L. Epstein and D. Schneider, IID: independently and indistingguishably distributed, Journal of Economic Theory, 113 (2003), 32-50. doi: 10.1016/S0022-0531(03)00121-2. [9] P. Huper, The use of Choquet capacities in statistics, Bulletin of the International Statistical Institute, 45 (1973), 181-191. [10] F. Maccheroni and M. Marinacci, A strong law of large number for capacities, Annals of Probability, 33 (2005), 1171-1178. doi: 10.1214/009117904000001062. [11] 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. [12] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy sets to Systems Analysis, Birkhauser,Basel, 1975. doi: 10.1007/978-3-0348-5921-9. [13] S. Peng, BSDE and related g-expectation, Pitman Research Notes in Mathematics Series, 364 (1997), 141-159. [14] S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type, Probability Theory and Related Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214. [15] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applicatios, Springer, Berlin Heidelberg, 2 (2007), 541-567. doi: 10.1007/978-3-540-70847-6_25. [16] S. Peng, Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, Stochastic Processes and their Applications, 118 (2008), 2223-2253. doi: 10.1016/j.spa.2007.10.015. [17] 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. [18] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint, arXiv:1002.4546. [19] M. L. Puri and D. A. Ralescu, Strong law of large numbers with respect to a set-valued probability mesure, The Annals of Probability, 11 (1983), 1051-1054. doi: 10.1214/aop/1176993455. [20] M. L. Puri and D. A. Ralescu, Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114 (1986), 409-422. doi: 10.1016/0022-247X(86)90093-4. [21] D. A. Ralescu, Radom-Nikodym theorem for fuzzy set-valued measures, Fuzzy Sets Theory and Applications, 177 (1986), 39-50. [22] D. A. Ralescu, Strong law of large numbers with respect to a fuzzy probability measure, Metron, 71 (2013), 201-206. doi: 10.1007/s40300-013-0022-z. [23] L. Wasserman and J. Kadane, Bayes's theorem for Choquent capacities, The Annals of Statistics, 18 (1990), 1328-1339. doi: 10.1214/aos/1176347752.

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##### References:
 [1] Z. Artsteun, Set-valued measures, Transactions of the American Mathematical Society, 165 (1972), 103-125. doi: 10.1090/S0002-9947-1972-0293054-4. [2] Z. Chen and L. Epstein, Ambiguity, risk and asset returns in continuous time, Econometrica, 70 (2002), 1403-1443. doi: 10.1111/1468-0262.00337. [3] Z. Chen and P. Wu, Strong laws of large numbers for Bernoulli experiments under ambiguity, Advances in Intelligent and Soft Computing, 100 (2011), 19-30. doi: 10.1007/978-3-642-22833-9_2. [4] Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilies, International Journal of Approximate Reasoning, 54 (2013), 365-377. doi: 10.1016/j.ijar.2012.06.002. [5] G. Cooman and E. Miranda, Weak and strong laws of large numbers for coherent lower precision, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432. doi: 10.1016/j.jspi.2007.10.020. [6] G. Debereu and D. Schmeidler, The Radom-Nikodym derivative of a correspondence, Proc. Sixth Berkeley Sympo. Math. Statist. Probab, Univ. of California Press, 2 (1970), 41-56. [7] L. DeRobertis and J. A. Hartigan, Bayesian inference using intervals of measures, Annals of Statistics, 9 (1981), 235-244. doi: 10.1214/aos/1176345391. [8] L. Epstein and D. Schneider, IID: independently and indistingguishably distributed, Journal of Economic Theory, 113 (2003), 32-50. doi: 10.1016/S0022-0531(03)00121-2. [9] P. Huper, The use of Choquet capacities in statistics, Bulletin of the International Statistical Institute, 45 (1973), 181-191. [10] F. Maccheroni and M. Marinacci, A strong law of large number for capacities, Annals of Probability, 33 (2005), 1171-1178. doi: 10.1214/009117904000001062. [11] 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. [12] C. V. Negoita and D. A. Ralescu, Applications of Fuzzy sets to Systems Analysis, Birkhauser,Basel, 1975. doi: 10.1007/978-3-0348-5921-9. [13] S. Peng, BSDE and related g-expectation, Pitman Research Notes in Mathematics Series, 364 (1997), 141-159. [14] S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type, Probability Theory and Related Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214. [15] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applicatios, Springer, Berlin Heidelberg, 2 (2007), 541-567. doi: 10.1007/978-3-540-70847-6_25. [16] S. Peng, Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation, Stochastic Processes and their Applications, 118 (2008), 2223-2253. doi: 10.1016/j.spa.2007.10.015. [17] 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. [18] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint, arXiv:1002.4546. [19] M. L. Puri and D. A. Ralescu, Strong law of large numbers with respect to a set-valued probability mesure, The Annals of Probability, 11 (1983), 1051-1054. doi: 10.1214/aop/1176993455. [20] M. L. Puri and D. A. Ralescu, Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114 (1986), 409-422. doi: 10.1016/0022-247X(86)90093-4. [21] D. A. Ralescu, Radom-Nikodym theorem for fuzzy set-valued measures, Fuzzy Sets Theory and Applications, 177 (1986), 39-50. [22] D. A. Ralescu, Strong law of large numbers with respect to a fuzzy probability measure, Metron, 71 (2013), 201-206. doi: 10.1007/s40300-013-0022-z. [23] L. Wasserman and J. Kadane, Bayes's theorem for Choquent capacities, The Annals of Statistics, 18 (1990), 1328-1339. doi: 10.1214/aos/1176347752.
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