    January  2019, 4: 4 doi: 10.1186/s41546-019-0038-2

## Law of large numbers and central limit theorem under nonlinear expectations

 Institute of Mathematics, Shandong University, Jinan 250100, Shandong Province, China

Received  November 19, 2018 Revised  April 2019 Published  April 2019

The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence $\left\{ {1/\sqrt n \left( {{X_1} + \cdots + {X_n}} \right)} \right\}_{i = 1}^\infty$ converges in law to a nonlinear normal distribution, called G-normal distribution, where $\left\{ {{X_i}} \right\}_{i = 1}^\infty$ is an i.i.d. sequence under the sublinear expectation. It's known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1).
Citation: Shige Peng. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 4-. doi: 10.1186/s41546-019-0038-2
##### References:
  Cabre, X. and Caffarelli, L.A. (1997). Fully nonlinear elliptic partial differential equations, American Mathematical Society. Caffarelli, L.A. (1989). Interior estimates for fully nonlinear equations, Ann. of Math. 130, 189-213. Peng, S. (2004). Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims, Acta Mathematicae Applicatae Sinica, Engl. Ser. 20, no. 2, 1-24. Peng, S. (2005). Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math. 26B, no. 2, 159-184. Peng, S. (2007). G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô's type. in Stochastic Analysis and Applications, The Abel Symposium 2005, Abel Symposia2, Edit. Benth et. al., 541-567, Springer-Verlag. Peng, S. (2008). Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under GExpectation. Stochastic Processes and their Applications 118(12), 2223-2253. Wang, L. (1992). On the regularity of fully nonlinear parabolic equations:II, Comm. Pure Appl. Math. 45, 141-178. show all references

##### References:
  Cabre, X. and Caffarelli, L.A. (1997). Fully nonlinear elliptic partial differential equations, American Mathematical Society. Caffarelli, L.A. (1989). Interior estimates for fully nonlinear equations, Ann. of Math. 130, 189-213. Peng, S. (2004). Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims, Acta Mathematicae Applicatae Sinica, Engl. Ser. 20, no. 2, 1-24. Peng, S. (2005). Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math. 26B, no. 2, 159-184. Peng, S. (2007). G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô's type. in Stochastic Analysis and Applications, The Abel Symposium 2005, Abel Symposia2, Edit. Benth et. al., 541-567, Springer-Verlag. Peng, S. (2008). Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under GExpectation. Stochastic Processes and their Applications 118(12), 2223-2253. Wang, L. (1992). On the regularity of fully nonlinear parabolic equations:II, Comm. Pure Appl. 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