A New Central Limit Theorem under Sublinear Expectations

By Shige Peng. 2008

228 citations recorded by the source corpus.

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.

Subfields: Finance, Management Science and Operations Research, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/cae50479c151c483e0316ccf8c0708c3ec52a870

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Citations in the corpus

Backward stochastic differential equations driven by G-Brownian motion (2012, background): “More details of this section can be found in [17], [18], [19], [20], [22].”
Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations (2016, methodology): “…of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng (2008b).”
Strong laws of large numbers for capacities (2010, background): “Recently, motivated by the risk measures, super-hedge pricing and modelling uncertainty in finance, Peng [11,12,13,14,15] initiated the notion of independent and identically distributed (IID) random variables under sub-linear expectations.”
G-Lévy processes under sublinear expectations (2009, methodology): “Recently a new central limit theorem (CLT) under sublinear expectations has been obtained in [24] based on a new i.”
Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm (2014, methodology): “Roughly speaking, a sub-linear expectation is related to a group of unknown linear expectations and the distribution under a sub-linear expectation is related to a group of probabilities (c.f. Lemma 2.4 of Peng (2008b)).”
Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm (2015, methodology): “Roughly speaking, a sub-linear expectation is related to a group of unknown linear expectations and the distribution under a sub-linear expectation is related to a group of probabilities (c.f. Lemma 2.4 of Peng (2008b)).”
Normal approximation by Stein’s method under sublinear expectations (2017, background): “05 38 4v 1 [ m at h. PR ] 1 5 N ov Peng (2008)([12]) proved the Central Limit Theorem under a sublinear expectation: Let (Xi)i≥1 be a sequence of i.i.d random variables under a sublinear expectation Ê with Ê[X1] = Ê[−X1] = 0 and Ê[|X1”
Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010, methodology): “The results of this chapter for G-Brownian motions were mainly obtained by Denis, Hu and Peng (2008) [37] (see also Denis and Martini (2006) [38] and the related comments after Chapter III).”
Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications (2014, methodology): “lds for independent and identically distributed random variables under a continuous sub-linear expectation if and only if the the corresponding Choquet integral is ﬁnite. We use the notations of Peng [14]. Let (Ω,F) be a given measurable space and let H be a linear space of real functions deﬁned on ”