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Limit theorems for non-additive probabilities or non-linear expectations arechallenging issues which have raised progressive interest recently. The purposeof this paper is to study the strong law of large numbers and the law of theiterated logarithm for a sequence of random variables in a sub-linearexpectation space under a concept of extended independence which is much weakerand easier to verify than the independence proposed by Peng (2008b). Weintroduce a concept of extended negative dependence which is an extension ofthis kind of weak independence and the extended negative independence relativeto classical probability appeared in recent literatures. Powerful tools as themoment inequality and Kolmogorov's exponential inequality are established forthis kind of extended negatively independent random variables, which improvethose of Chen, Chen and Ng(2010) a lot. And the strong law of large numbers andthe law of iterated logarithm are obtained by applying these inequalities.

Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations