Law of large numbers and central limit theorem under nonlinear expectations

Shige Peng

doi:10.1186/s41546-019-0038-2

Article Contents

2019, Volume 4: 4. Doi: 10.1186/s41546-019-0038-2

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Law of large numbers and central limit theorem under nonlinear expectations

Shige Peng^,

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

Shige Peng, peng@sdu.edu.cn

Received: November 18, 2018

Revised: March 31, 2019

Published: July 2019

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Abstract

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).

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References

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