In this paper, with a new notion of exponential independence for random variables under an upper expectation, we establish a kind of strong laws of large numbers for capacities. Our limit theorems show that the cluster points of empirical averages not only lie in the interval between the upper expectation and the lower expectation with lower probability one, but such an interval also is the unique smallest interval of all intervals in which the limit points of empirical averages lie with lower probability one. Furthermore, we also show that the cluster points of empirical averages could reach the upper expectation and lower expectation with upper probability one.

In this paper, we obtain the equivalent relations between Kolmogorov maximal inequality and Hájek-Rényi maximal inequality both in moment and capacity types in sublinear expectation spaces. Based on these, we establish several strong laws of large numbers for general random variables and obtain the growth rate of the partial sums. In a first application, a strong law of large numbers for negatively dependent random variables is obtained. In a second application, we consider the normalizing sequence {log n} n≥1 and get some special limit properties in sublinear expectation spaces.

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This paper introduces new notions of Fubini independence and Exponential independence of random variables under capacities to fit Ellsberg's model, and finds out the relations between Fubini independence, Exponential independence, MacCheroni and Marinacci's independence and Peng's independence. As an application, we give a weak law of large numbers for capacities under Exponential independence. Simulations show that Ellsberg's model enjoy the weak law of large numbers when there is mean uncertainty with or without variance uncertainty.