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

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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

Zengjing Chen ^1, , Yuting Lan ^1, and Gaofeng Zong ^2,

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

Abstract
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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.

Keywords: upper-lower set-valued probability, Set-valued probability, non-additive probability, fuzzy-set valued probability, strong law of large numbers..

Mathematics Subject Classification: Primary: 60F15; Secondary: 52A22, 60B1.

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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