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Two laws of large numbers for sublinear expectations  [PDF]
Chuanfeng Sun
Mathematics , 2015,
Abstract: In this paper, we consider the sublinear expectation on bounded random vari- ables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the no- tion of independence for random variable sequences and regular property for sublinear expectations, we get a strong one.
Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations  [PDF]
Ze-Chun Hu,Ling Zhou
Mathematics , 2012,
Abstract: In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.
On a General Approach to the Strong Laws of Large Numbers  [PDF]
István Fazekas
Mathematics , 2014,
Abstract: A general method to obtain strong laws of large numbers is studied. The method is based on abstract H\'ajek-R\'enyi type maximal inequalities. The rate of convergence in the law of large numbers is also considered. Some applications for weakly dependent sequences are given.
Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces  [PDF]
Samuel N. Cohen
Mathematics , 2011,
Abstract: We consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering a decomposition of the space in terms of the supports of the measures representing our sublinear expectation, we give a simple construction, in a quasi-sure sense, of the (linear) conditional expectations, and hence give a representation for the conditional sublinear expectation. We also show an aggregation property holds, and give an equivalence between consistency and a pasting property of measures.
Constructing Sublinear Expectations on Path Space  [PDF]
Marcel Nutz,Ramon van Handel
Mathematics , 2012, DOI: 10.1016/j.spa.2013.03.022
Abstract: We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.
Sublinear Expectations and Martingales in discrete time  [PDF]
Samuel Cohen,Shaolin Ji,Shige Peng
Mathematics , 2011,
Abstract: We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of martingales, and martingale convergence. We also give a theory of BSDEs in the context of sublinear expectations and a finite-state space, including general existence and comparison results.
Some inequalities and limit theorems under sublinear expectations  [PDF]
Ze-Chun Hu,Yan-Zhi Yang
Mathematics , 2012,
Abstract: 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 special version of Kolmogrov's law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.
On general strong laws of large numbers for fields of random variables  [PDF]
Cheikhna Hamallah,Gane Samb Lo
Mathematics , 2011,
Abstract: A general method to prove strong laws of large numbers for random fields is given. It is based on the H\'ajek - R\'enyi type method presented in Nosz\'aly and T\'om\'acs \cite{noszaly} and in T\'om\'acs and L\'ibor \cite{thomas06}. Nosz\'aly and T\'om\'acs \cite{noszaly} obtained a H\'ajek-R\'enyi type maximal inequality for random fields using moments inequalities. Recently, T\'om\'acs and L\'ibor \cite{thomas06} obtained a H\'ajek-R\'enyi type maximal inequality for random sequences based on probabilities, but not for random fields. In this paper we present a H\'ajek-R\'enyi type maximal inequality for random fields, using probabilities, which is an extension of the main results of Nosz\'aly and T\'om\'acs \cite{noszaly} by replacing moments by probabilities and a generalization of the main results of T\'om\'acs and L\'ibor \cite% {thomas06} for random sequences to random fields. We apply our results to establishing a logarithmically weighted sums without moment assumptions and under general dependence conditions for random fields.
The Independence under Sublinear Expectations  [PDF]
Mingshang Hu
Mathematics , 2011,
Abstract: We show that, for two non-trivial random variables X and Y under a sublinear expectation space, if X is independent from Y and Y is independent from X, then X and Y must be maximally distributed.
A central limit theorem under sublinear expectations  [PDF]
Min Li,Yufeng Shi
Mathematics , 2010, DOI: 10.1007/s11425-010-3156-y
Abstract: In this paper we consider a sequence of random variables with mean uncertainty in a sublinear expectation space. Without the hypothesis of identical distributions, we show a new central limit theorem under the sublinear expectations.
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