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

Previous Article
Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process
MCRF Home
This Issue
Next Article
BMO martingales and positive solutions of heat equations

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
Related Papers

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 & 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. doi: 10.1090/S0002-9947-1972-0293054-4. Google Scholar
[2]	Z. Chen and L. Epstein, Ambiguity, risk and asset returns in continuous time,, Econometrica, 70 (2002), 1403. doi: 10.1111/1468-0262.00337. Google Scholar
[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. doi: 10.1007/978-3-642-22833-9_2. Google Scholar
[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. doi: 10.1016/j.ijar.2012.06.002. Google Scholar
[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. doi: 10.1016/j.jspi.2007.10.020. Google Scholar
[6]	G. Debereu and D. Schmeidler, The Radom-Nikodym derivative of a correspondence,, Proc. Sixth Berkeley Sympo. Math. Statist. Probab, 2 (1970), 41. Google Scholar
[7]	L. DeRobertis and J. A. Hartigan, Bayesian inference using intervals of measures,, Annals of Statistics, 9 (1981), 235. doi: 10.1214/aos/1176345391. Google Scholar
[8]	L. Epstein and D. Schneider, IID: independently and indistingguishably distributed,, Journal of Economic Theory, 113 (2003), 32. doi: 10.1016/S0022-0531(03)00121-2. Google Scholar
[9]	P. Huper, The use of Choquet capacities in statistics,, Bulletin of the International Statistical Institute, 45 (1973), 181. Google Scholar
[10]	F. Maccheroni and M. Marinacci, A strong law of large number for capacities,, Annals of Probability, 33 (2005), 1171. doi: 10.1214/009117904000001062. Google Scholar
[11]	M. Marinacci, Limit laws for non-additive probabilities and their frequentist interpretation,, Journal of Economic Theory, 84 (1999), 145. doi: 10.1006/jeth.1998.2479. Google Scholar
[12]	C. V. Negoita and D. A. Ralescu, Applications of Fuzzy sets to Systems Analysis,, Birkhauser, (1975). doi: 10.1007/978-3-0348-5921-9. Google Scholar
[13]	S. Peng, BSDE and related g-expectation,, Pitman Research Notes in Mathematics Series, 364 (1997), 141. Google Scholar
[14]	S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type,, Probability Theory and Related Fields, 113 (1999), 473. doi: 10.1007/s004400050214. Google Scholar
[15]	S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type,, Stochastic Analysis and Applicatios, 2 (2007), 541. doi: 10.1007/978-3-540-70847-6_25. Google Scholar
[16]	S. Peng, Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation,, Stochastic Processes and their Applications, 118 (2008), 2223. doi: 10.1016/j.spa.2007.10.015. Google Scholar
[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. doi: 10.1007/s11425-009-0121-8. Google Scholar
[18]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint,, , (). Google Scholar
[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. doi: 10.1214/aop/1176993455. Google Scholar
[20]	M. L. Puri and D. A. Ralescu, Fuzzy random variables., Journal of Mathematical Analysis and Applications, 114 (1986), 409. doi: 10.1016/0022-247X(86)90093-4. Google Scholar
[21]	D. A. Ralescu, Radom-Nikodym theorem for fuzzy set-valued measures,, Fuzzy Sets Theory and Applications, 177 (1986), 39. Google Scholar
[22]	D. A. Ralescu, Strong law of large numbers with respect to a fuzzy probability measure,, Metron, 71 (2013), 201. doi: 10.1007/s40300-013-0022-z. Google Scholar
[23]	L. Wasserman and J. Kadane, Bayes's theorem for Choquent capacities,, The Annals of Statistics, 18 (1990), 1328. doi: 10.1214/aos/1176347752. Google Scholar

show all references

References:

[1]	Z. Artsteun, Set-valued measures,, Transactions of the American Mathematical Society, 165 (1972), 103. doi: 10.1090/S0002-9947-1972-0293054-4. Google Scholar
[2]	Z. Chen and L. Epstein, Ambiguity, risk and asset returns in continuous time,, Econometrica, 70 (2002), 1403. doi: 10.1111/1468-0262.00337. Google Scholar
[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. doi: 10.1007/978-3-642-22833-9_2. Google Scholar
[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. doi: 10.1016/j.ijar.2012.06.002. Google Scholar
[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. doi: 10.1016/j.jspi.2007.10.020. Google Scholar
[6]	G. Debereu and D. Schmeidler, The Radom-Nikodym derivative of a correspondence,, Proc. Sixth Berkeley Sympo. Math. Statist. Probab, 2 (1970), 41. Google Scholar
[7]	L. DeRobertis and J. A. Hartigan, Bayesian inference using intervals of measures,, Annals of Statistics, 9 (1981), 235. doi: 10.1214/aos/1176345391. Google Scholar
[8]	L. Epstein and D. Schneider, IID: independently and indistingguishably distributed,, Journal of Economic Theory, 113 (2003), 32. doi: 10.1016/S0022-0531(03)00121-2. Google Scholar
[9]	P. Huper, The use of Choquet capacities in statistics,, Bulletin of the International Statistical Institute, 45 (1973), 181. Google Scholar
[10]	F. Maccheroni and M. Marinacci, A strong law of large number for capacities,, Annals of Probability, 33 (2005), 1171. doi: 10.1214/009117904000001062. Google Scholar
[11]	M. Marinacci, Limit laws for non-additive probabilities and their frequentist interpretation,, Journal of Economic Theory, 84 (1999), 145. doi: 10.1006/jeth.1998.2479. Google Scholar
[12]	C. V. Negoita and D. A. Ralescu, Applications of Fuzzy sets to Systems Analysis,, Birkhauser, (1975). doi: 10.1007/978-3-0348-5921-9. Google Scholar
[13]	S. Peng, BSDE and related g-expectation,, Pitman Research Notes in Mathematics Series, 364 (1997), 141. Google Scholar
[14]	S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type,, Probability Theory and Related Fields, 113 (1999), 473. doi: 10.1007/s004400050214. Google Scholar
[15]	S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type,, Stochastic Analysis and Applicatios, 2 (2007), 541. doi: 10.1007/978-3-540-70847-6_25. Google Scholar
[16]	S. Peng, Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation,, Stochastic Processes and their Applications, 118 (2008), 2223. doi: 10.1016/j.spa.2007.10.015. Google Scholar
[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. doi: 10.1007/s11425-009-0121-8. Google Scholar
[18]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, preprint,, , (). Google Scholar
[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. doi: 10.1214/aop/1176993455. Google Scholar
[20]	M. L. Puri and D. A. Ralescu, Fuzzy random variables., Journal of Mathematical Analysis and Applications, 114 (1986), 409. doi: 10.1016/0022-247X(86)90093-4. Google Scholar
[21]	D. A. Ralescu, Radom-Nikodym theorem for fuzzy set-valued measures,, Fuzzy Sets Theory and Applications, 177 (1986), 39. Google Scholar
[22]	D. A. Ralescu, Strong law of large numbers with respect to a fuzzy probability measure,, Metron, 71 (2013), 201. doi: 10.1007/s40300-013-0022-z. Google Scholar
[23]	L. Wasserman and J. Kadane, Bayes's theorem for Choquent capacities,, The Annals of Statistics, 18 (1990), 1328. doi: 10.1214/aos/1176347752. Google Scholar

[1]	Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[2]	Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[3]	Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087
[4]	Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548
[5]	Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[6]	Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
[7]	Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246
[8]	Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[9]	Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309
[10]	Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[11]	Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051
[12]	C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[13]	Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567
[14]	Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35
[15]	Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019
[16]	Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745
[17]	Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041
[18]	Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57
[19]	Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[20]	Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences