Strong Limit Theorems for Extended Independent Random Variables and Extended Negatively Dependent Random Variables under Sub-Linear Expectations

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have attracted a lot of interest recently. The purpose 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 [20]. We introduce a concept of extended negative dependence which is an extension of the kind of weak independence and the extended negative independence relative to classical probability that has appeared in the recent literature. Powerful tools such as moment inequality and Kolmogorov's exponential inequality are established for these kinds of extended negatively independent random variables, and these tools improve a lot upon those of Chen, Chen and Ng [1]. The strong law of large numbers and the law of iterated logarithm are also obtained by applying these inequalities.