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Article Contents

# Extension of the strong law of large numbers for capacities

• * Corresponding author: Panyu Wu
This research is supported by Taishan Scholars Project and the National Natural Science Foundation of China (Grants 11231005, 11601280, 11471190, 11701331 and 11871050), the Natural Science Foundation of Shandong Province of China (Grants ZR2016AQ11 and ZR2016AQ13).
• 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.

Mathematics Subject Classification: Primary: 60F15; Secondary: 28A12.

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