Abstract
We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen Z J, Epstein L. Ambiguity, risk and asset returns in continuous time. Econometrica, 2002, 70: 1403–1443
Chen Z J, Wu P Y, Li B M. A strong law of large numbers for non-additive probabilities. Internat J Approx Reason, 2013, 54: 365–377
Dow J, Werlang S R C. Laws of large numbers for non-additive probabilities. Http://hdl.handle.net/10438/727, 1993
Epstein L, Schneider D. IID: Independently and indistinguishably distributed. J Econom Theory, 2003, 113: 32–50
Feynman R, Leighton R, Sands M. The Feynman Lectures on Physics, Volume 3: Quantum Mechanics. Reading: Addison-Wesley, 1963
Gilboa I. Expected utility theory with purely subjective non-additive probabilities. J Math Econom, 1987, 16: 65–68
Huber P J. The use of Choquet capacities in statistics. Bull Inst Internat Statist, 1973, 45: 181–191
Maccheroni F, Marinacci M. A strong law of large numbers for capacities. Ann Probab, 2005, 33: 1171–1178
Marinacci M. Limit laws for non-additive probabilities and their frequentist interpretation. J Econom Theory, 1999, 84: 145–195
Peng S G. BSDE and related g-expectation. Pitman Res Notes Math Ser, 1997, 364: 141–159
Peng S G. Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type. Probab Theory Related Fields, 1999, 113: 473–499
Peng S G. Nonlinear expectations and nonlinear Markov chains. Chin Ann Math Ser B, 2005, 26: 159–184
Peng S G. G-expectation, G-Brownian motion and related stochastic calculus of Ito type. Stoch Anal Appl, 2007, 2: 541–567
Peng S G. Law of large numbers and central limit theorem under nonlinear expectations. ArXiv:math.PR/0702358vl, 2007
Peng S G. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl, 2008, 118: 2223–2253
Peng S G. A new central limit theorem under sublinear expectations. ArXiv:0803.2656v1, 2008
Peng S G. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci China Ser A, 2009, 52: 1391–1411
Schmeidler D. Subjective probability and expected utility without additivity. Econometrica, 1989, 57: 571–587
Wakker P. Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle. Econometrica, 2001, 69: 1039–1059
Walley P, Fine T L. Towards a frequentist theory of upper and lower probability. Ann Statist, 1982, 10: 741–761
Wasserman L, Kadane J. Bayes’s theorem for Choquet capacities. Ann Statist, 1990, 18: 1328–1339
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, Z. Strong laws of large numbers for sub-linear expectations. Sci. China Math. 59, 945–954 (2016). https://doi.org/10.1007/s11425-015-5095-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-015-5095-0