    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

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. Google Scholar  Caffarelli, L.A. (1989). Interior estimates for fully nonlinear equations, Ann. of Math. 130, 189-213. Google Scholar  Peng, S. (2004). Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims, Acta Mathematicae Applicatae Sinica, Engl. Ser. 20, no. 2, 1-24. Google Scholar  Peng, S. (2005). Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math. 26B, no. 2, 159-184. Google Scholar  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. Google Scholar  Peng, S. (2008). Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under GExpectation. Stochastic Processes and their Applications 118(12), 2223-2253. Google Scholar  Wang, L. (1992). On the regularity of fully nonlinear parabolic equations:II, Comm. Pure Appl. Math. 45, 141-178. Google Scholar

show all references

##### References:
  Cabre, X. and Caffarelli, L.A. (1997). Fully nonlinear elliptic partial differential equations, American Mathematical Society. Google Scholar  Caffarelli, L.A. (1989). Interior estimates for fully nonlinear equations, Ann. of Math. 130, 189-213. Google Scholar  Peng, S. (2004). Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims, Acta Mathematicae Applicatae Sinica, Engl. Ser. 20, no. 2, 1-24. Google Scholar  Peng, S. (2005). Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math. 26B, no. 2, 159-184. Google Scholar  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. Google Scholar  Peng, S. (2008). Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under GExpectation. Stochastic Processes and their Applications 118(12), 2223-2253. Google Scholar  Wang, L. (1992). On the regularity of fully nonlinear parabolic equations:II, Comm. Pure Appl. Math. 45, 141-178. Google Scholar
  Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597  James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167  Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477  Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143  H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5 (4) : 647-667. doi: 10.3934/mbe.2008.5.647  Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497  Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012  David Simmons. Conditional measures and conditional expectation; Rohlin's Disintegration Theorem. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2565-2582. doi: 10.3934/dcds.2012.32.2565  Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270  . Publisher Correction to: Probability, uncertainty and quantitative risk, volume 4. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 7-. doi: 10.1186/s41546-019-0041-7  Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101  Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks & Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561  Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639  H.Thomas Banks, Shuhua Hu. Nonlinear stochastic Markov processes and modeling uncertainty in populations. Mathematical Biosciences & Engineering, 2012, 9 (1) : 1-25. doi: 10.3934/mbe.2012.9.1  Tatiana Filippova. Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty. Conference Publications, 2011, 2011 (Special) : 410-419. doi: 10.3934/proc.2011.2011.410  Nguyen Dinh Cong, Doan Thai Son, Stefan Siegmund, Hoang The Tuan. An instability theorem for nonlinear fractional differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3079-3090. doi: 10.3934/dcdsb.2017164  Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692  Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081  Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711  Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048

Impact Factor: