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A central limit theorem for sets of probability measures

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  • Chen, Zengjing
  • Epstein, Larry G.

Abstract

We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of (suitably equivalent) probability measures. The limit is defined by a backward stochastic differential equation that can be interpreted as modeling an ambiguous continuous-time random walk.

Suggested Citation

  • Chen, Zengjing & Epstein, Larry G., 2022. "A central limit theorem for sets of probability measures," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 424-451.
    • Handle: RePEc:eee:spapps:v:152:y:2022:i:c:p:424-451
    DOI: 10.1016/j.spa.2022.07.003
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    References listed on IDEAS

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    1. Tamer, Elie, 2010. "Partial Identification in Econometrics," Scholarly Articles 34728615, Harvard University Department of Economics.
    2. Larry G. Epstein & Martin Schneider, 2010. "Ambiguity and Asset Markets," Annual Review of Financial Economics, Annual Reviews, vol. 2(1), pages 315-346, December.
    3. Gilboa,Itzhak, 2009. "Theory of Decision under Uncertainty," Cambridge Books, Cambridge University Press, number 9780521517324.
    4. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    5. Chen, Zengjing & Peng, Shige, 2000. "A general downcrossing inequality for g-martingales," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 169-175, January.
    6. Larry G. Epstein & Hiroaki Kaido & Kyoungwon Seo, 2016. "Robust Confidence Regions for Incomplete Models," Econometrica, Econometric Society, vol. 84, pages 1799-1838, September.
    7. Garud N. Iyengar, 2005. "Robust Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 257-280, May.
    8. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    9. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    10. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    11. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    12. Epstein, Larry G. & Schneider, Martin, 2003. "IID: independently and indistinguishably distributed," Journal of Economic Theory, Elsevier, vol. 113(1), pages 32-50, November.
    13. Chen, Pingyan & Sung, Soo Hak, 2016. "On the strong laws of large numbers for weighted sums of random variables," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 87-93.
    14. Elie Tamer, 2010. "Partial Identification in Econometrics," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 167-195, September.
    15. Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
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    Cited by:

    1. Zengjing Chen & Larry G. Epstein & Guodong Zhang, 2022. "Approximate optimality and the risk/reward tradeoff in a class of bandit problems," Papers 2210.08077, arXiv.org, revised Dec 2023.
    2. Chen, Zengjing & Epstein, Larry G. & Zhang, Guodong, 2023. "A central limit theorem, loss aversion and multi-armed bandits," Journal of Economic Theory, Elsevier, vol. 209(C).
    3. Zengjing Chen & Huaijin Liang & Wei Wang & Xiaodong Yan, 2022. "Long bet will lose: demystifying seemingly fair gambling via two-armed Futurity bandit," Papers 2212.11766, arXiv.org.

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