Skip to main content
SpringerLink
Log in

Strong laws of large numbers for sub-linear expectations

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen Z J, Epstein L. Ambiguity, risk and asset returns in continuous time. Econometrica, 2002, 70: 1403–1443

    Article  MathSciNet  MATH  Google Scholar 

  2. 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

    Article  MathSciNet  MATH  Google Scholar 

  3. Dow J, Werlang S R C. Laws of large numbers for non-additive probabilities. Http://hdl.handle.net/10438/727, 1993

    Google Scholar 

  4. Epstein L, Schneider D. IID: Independently and indistinguishably distributed. J Econom Theory, 2003, 113: 32–50

    Article  MathSciNet  MATH  Google Scholar 

  5. Feynman R, Leighton R, Sands M. The Feynman Lectures on Physics, Volume 3: Quantum Mechanics. Reading: Addison-Wesley, 1963

    MATH  Google Scholar 

  6. Gilboa I. Expected utility theory with purely subjective non-additive probabilities. J Math Econom, 1987, 16: 65–68

    Article  MathSciNet  MATH  Google Scholar 

  7. Huber P J. The use of Choquet capacities in statistics. Bull Inst Internat Statist, 1973, 45: 181–191

    MathSciNet  Google Scholar 

  8. Maccheroni F, Marinacci M. A strong law of large numbers for capacities. Ann Probab, 2005, 33: 1171–1178

    Article  MathSciNet  MATH  Google Scholar 

  9. Marinacci M. Limit laws for non-additive probabilities and their frequentist interpretation. J Econom Theory, 1999, 84: 145–195

    Article  MathSciNet  MATH  Google Scholar 

  10. Peng S G. BSDE and related g-expectation. Pitman Res Notes Math Ser, 1997, 364: 141–159

    MATH  Google Scholar 

  11. Peng S G. Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer type. Probab Theory Related Fields, 1999, 113: 473–499

    Article  MathSciNet  MATH  Google Scholar 

  12. Peng S G. Nonlinear expectations and nonlinear Markov chains. Chin Ann Math Ser B, 2005, 26: 159–184

    Article  MathSciNet  MATH  Google Scholar 

  13. Peng S G. G-expectation, G-Brownian motion and related stochastic calculus of Ito type. Stoch Anal Appl, 2007, 2: 541–567

    MathSciNet  MATH  Google Scholar 

  14. Peng S G. Law of large numbers and central limit theorem under nonlinear expectations. ArXiv:math.PR/0702358vl, 2007

    Google Scholar 

  15. Peng S G. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl, 2008, 118: 2223–2253

    Article  MathSciNet  MATH  Google Scholar 

  16. Peng S G. A new central limit theorem under sublinear expectations. ArXiv:0803.2656v1, 2008

    Google Scholar 

  17. 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

    Article  MathSciNet  MATH  Google Scholar 

  18. Schmeidler D. Subjective probability and expected utility without additivity. Econometrica, 1989, 57: 571–587

    Article  MathSciNet  MATH  Google Scholar 

  19. Wakker P. Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle. Econometrica, 2001, 69: 1039–1059

    Article  MathSciNet  MATH  Google Scholar 

  20. Walley P, Fine T L. Towards a frequentist theory of upper and lower probability. Ann Statist, 1982, 10: 741–761

    Article  MathSciNet  MATH  Google Scholar 

  21. Wasserman L, Kadane J. Bayes’s theorem for Choquet capacities. Ann Statist, 1990, 18: 1328–1339

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    ZengJing Chen

Authors
  1. ZengJing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ZengJing Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-5095-0

Keywords

MSC(2010)

Search

Navigation