• Corpus ID: 15546914

# A New Central Limit Theorem under Sublinear Expectations

@article{Peng2008ANC,
title={A New Central Limit Theorem under Sublinear Expectations},
author={Shige Peng},
journal={arXiv: Probability},
year={2008}
}
• S. Peng
• Published 18 March 2008
• Mathematics
• arXiv: Probability
We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.
162 Citations

### On the functional central limit theorem with mean-uncertainty

We introduce a new basic model for independent and identical distributed sequence on the canonical space (RN,B(RN)) via probability kernels with model uncertainty. Thanks to the well-defined upper

### Central limit theorems for sub-linear expectation under the Lindeberg condition

• Cheng Hu
• Mathematics
Journal of inequalities and applications
• 2018
The central limit theorems for sub-linear expectation for a sequence of independent random variables without assumption of identical distribution are investigated and the central limit theorem for capacity for summability methods under the Lindeberg condition is obtained.

### A Weighted Central Limit Theorem Under Sublinear Expectations

• Mathematics
• 2011
In this article, we investigate a central limit theorem for weighted sum of independent random variables under sublinear expectations. It is turned out that our results are natural extensions of the

### An α -stable limit theorem under sublinear expectation

• Mathematics
• 2016
For α ∈ ( 1 , 2 ) , we present a generalized central limit theorem for α -stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for

### Nonlinear Expectations and Stochastic Calculus under Uncertainty

• S. Peng
• Mathematics
Probability Theory and Stochastic Modelling
• 2019
In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the

### Convergences of Random Variables Under Sublinear Expectations

• Mathematics
Chinese Annals of Mathematics, Series B
• 2018
In this note, the authors survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear

### An $\alpha$-stable limit theorem under sublinear expectation

• Mathematics
• 2014
For $\alpha\in (1,2)$, we present a generalized central limit theorem for $\alpha$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate

### Some inequalities and limit theorems under sublinear expectations

• Mathematics
• 2012
In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob’s inequality for submartingale and Kolmogrov’s inequality. By Kolmogrov’s inequality, we obtain a

### General laws of large numbers under sublinear expectations

• Mathematics
• 2011
ABSTRACT In this paper, under some weaker conditions, we give three laws of large numbers (LLNs) under sublinear expectations (capacities), which extend the LLN under sublinear expectations in Peng

### A law of the iterated logarithm under sublinear expectations

• Mathematics
• 2011
In this paper, with the notion of independent and identically distributed (IID) random variables under sublinear expectations initiated by Peng, we develop a law of the iterated logarithm (LIL) for

## References

SHOWING 1-10 OF 25 REFERENCES

### Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths

• Mathematics
• 2008
In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation—G-expectation. Many results can

### G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty

We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the

### Limit Laws for Non-additive Probabilities and Their Frequentist Interpretation

Abstract In this paper we prove several limit laws for non-additive probabilities. In particular, we prove that, under a multiplicative notion of independence and a regularity condition, if the

### NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

This paper deals with nonlinear expectations. The author obtains a nonlinear generalization of the well-known Kolmogorov's consistent theorem and then use it to construct filtration-consistent

### NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

This paper deals with nonlinear expectations. The author obtains a nonlinear generalization of the well-known Kolmogorov's consistent theorem and then use it to construct filtration-consistent

### G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type

We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution.

### A strong law of large numbers for capacities

• Mathematics
• 2005
We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the

### A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN THE PRESENCE OF MODEL UNCERTAINTY

• Mathematics
• 2006
The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a

### User’s guide to viscosity solutions of second order partial differential equations

• Mathematics
• 1992
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence