Bernoulli

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  • Volume 25, Number 4A (2019), 2564-2596.

Limit theorems with rate of convergence under sublinear expectations

Xiao Fang, Shige Peng, Qi-Man Shao, and Yongsheng Song

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Abstract

Under the sublinear expectation $\mathbb{E}[\cdot]:=\mathop{\mathrm{sup}}_{\theta\in\Theta}E_{\theta}[\cdot]$ for a given set of linear expectations $\{E_{\theta}:\theta\in\Theta\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the $G$-normal distribution, which was used as the limit in Peng’s (Law of large numbers and central limit theorem under nonlinear expectations (2007) Preprint) central limit theorem, in a probability space.

Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2564-2596.

Dates
Received: December 2017
Revised: May 2018
First available in Project Euclid: 13 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1568362036

Digital Object Identifier
doi:10.3150/18-BEJ1063

Mathematical Reviews number (MathSciNet)
MR4003558

Zentralblatt MATH identifier
07110105

Keywords
central limit theorem $G$-normal distribution law of large numbers rate of convergence Stein’s method sublinear expectation

Citation

Fang, Xiao; Peng, Shige; Shao, Qi-Man; Song, Yongsheng. Limit theorems with rate of convergence under sublinear expectations. Bernoulli 25 (2019), no. 4A, 2564--2596. doi:10.3150/18-BEJ1063. https://projecteuclid.org/euclid.bj/1568362036


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References

  • [1] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
    Zentralblatt MATH: 0980.91042
    Digital Object Identifier: doi:10.1111/1467-9965.00068
  • [2] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Heidelberg: Springer.
  • [3] Chen, Z. and Epstein, L. (2002). Ambiguity, risk, and asset returns in continuous time. Econometrica 70 1403–1443.
    Zentralblatt MATH: 1121.91359
    Digital Object Identifier: doi:10.1111/1468-0262.00337
  • [4] Delbaen, F., Peng, S. and Rosazza Gianin, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14 449–472.
    Zentralblatt MATH: 1226.91025
    Digital Object Identifier: doi:10.1007/s00780-009-0119-7
  • [5] Denis, L., Hu, M. and Peng, S. (2011). Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths. Potential Anal. 34 139–161.
    Zentralblatt MATH: 1225.60057
    Digital Object Identifier: doi:10.1007/s11118-010-9185-x
  • [6] Föllmer, H. and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time, extended ed. Berlin: de Gruyter.
  • [7] Huber, P.J. (1981). Robust Statistics. New York: Wiley.
    Zentralblatt MATH: 0536.62025
  • [8] Jin, H. and Peng, S. (2016). Optimal unbiased estimation for maximal distribution. Preprint. Available at http://arxiv.org/abs/1611.07994v1.
  • [9] Lieberman, G.M. (1996). Second Order Parabolic Differential Equations. River Edge, NJ: World Scientific.
    Zentralblatt MATH: 0884.35001
  • [10] Maccheroni, F. and Marinacci, M. (2005). A strong law of large numbers for capacities. Ann. Probab. 33 1171–1178.
    Zentralblatt MATH: 1074.60041
    Digital Object Identifier: doi:10.1214/009117904000001062
    Project Euclid: euclid.aop/1115386722
  • [11] Marinacci, M. (1999). Limit laws for non-additive probabilities and their frequentist interpretation. J. Econom. Theory 84 145–195.
    Zentralblatt MATH: 0921.90005
    Digital Object Identifier: doi:10.1006/jeth.1998.2479
  • [12] Peng, S. (1997). Backward SDE and related $g$-expectation. In Backward Stochastic Differential Equations (Paris, 1995–1996). Pitman Res. Notes Math. Ser. 364 141–159. Harlow: Longman.
  • [13] Peng, S. (2007). Law of large numbers and central limit theorem under nonlinear expectations. Preprint. Available at https://arxiv.org/abs/math/0702358.
  • [14] Peng, S. (2010). Nonlinear expectations and stochastic calculus under uncertainty. Preprint. Available at http://arxiv.org/abs/1002.4546v1.
  • [15] Röllin, A. (2018). On quantitative bounds in the mean martingale central limit theorem. Statist. Probab. Lett. 138 171–176.
  • [16] Song, Y. (2017). Normal approximation by Stein’s method under sublinear expectations. Preprint. Available at http://arxiv.org/abs/1711.05384.
  • [17] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971). Probability Theory II 583–602. Berkeley, CA: Univ. California Press.
  • [18] Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Monographs on Statistics and Applied Probability 42. London: Chapman & Hall.
    Zentralblatt MATH: 0732.62004

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