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.

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.

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.

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.

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.

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.

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.

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.

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.

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.