Strong Law of Large Numbers under an Upper Probability

Abstract

Strong law of large numbers is a fundamental theory in probability and statistics. When the measure tool is nonadditive, this law is very different from additive case. In 2010 Chen investigated the strong law of large numbers under upper probabilityVby assumingVis continuous. This assumption is very strong. Upper probabilities may not be continuous. In this paper we prove the strong law of large numbers for an upper probability without the continuity assumption whereby random variables are quasi-continuous and the upper probability is generated by a weakly compact family of probabilities on a complete and separable metric sample space.

Keywords

Strong Law of Large Numbers; Upper Probability; Weakly Compact; Independence; Quasi-Continuous

Strong Law of Large Numbers; Upper Probability; Weakly Compact; Independence; Quasi-Continuous

Cite this paper

X. Chen, "Strong Law of Large Numbers under an Upper Probability,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2056-2062. doi: 10.4236/am.2012.312A284.

X. Chen, "Strong Law of Large Numbers under an Upper Probability,"

References

[1] M. Marinacci, “Limit Laws of Non-Additive Probabilities and Their Frequentist Interpretation,” Journal of Economic Theory, Vol. 84, No. 2, 1999, pp. 145-195.
doi:10.1006/jeth.1998.2479

[2] F. Maccheroni and M. Marinacci, “A Strong Law of Large Numbers for Capacities,” Annals of Probability, Vol. 33, No. 3, 2005, pp. 1171-1178.
doi:10.1214/009117904000001062

[3] Z. Chen, “Strong Laws of Large Numbers for Capacities,” Unpublished, 2010. http://arxiv.org/abs/1006.0749

[4] S. Peng, “Nonlinear Expectations and Stochastic Calculus under Uncertainty-with Robust Central Limit Theorem and G-Brownian Motion,” Unpublished, 2012.
http://arxiv.org/abs/1002.4546

[5] Z. Chen, P. Wu and B. Li, “A Strong Law of Large Numbers for Non-additive Probabilities,” International Journal of Approximate Reasoning, 2012.
http://www.sciencedirect.com/science/article/pii/S0888613X12000783

[6] F. Hu, “General Laws of Large Numbers under Sublinear Expectations,” 2012. http://arxiv.org/abs/1104.5296

[7] Z. Chen and P. Wu, “Strong Laws of Large Numbers for Bernoulli Experiments under Ambiguity,” Nonlinear Mathematics for Uncertainty and Its Applications, Advances in Intelligent and Soft Computing, Vol. 100, 2011, pp. 19-30.
http://link.springer.com/chapter/10.1007%2F978-3-642-22833-9_2?LI=true

[8] J. Xu and B. Zhang, “Martingale Property and Capacity under G-Framework,” Electronic Journal of Probability, Vol. 15, No. 67, 2010, pp. 2041-2068.

[9] G. Choquet, “Theory of Capacities,” Annales de Institute Fourier, Vol. 5, 1954, pp. 131-295.

[10] P. J. Huber and V. Strassen, “Minimax Tests and the Neyman-Pearson Lemma for Capacities,” The Annals of Statistics, Vol. 1. No. 2, 1973, pp. 251-263.
doi:10.1214/aos/1176342363

[11] L. Denis, M. Hu and S. Peng, “Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths,” Potential Analysis, Vol. 34, No. 2, 2011, pp. 139-161.
doi:10.1007/s11118-010-9185-x