Abstract
ABSTRACTIn this paper, under some weaker conditions, we give three laws of large numbers (LLNs) under sublinear expectations (capacities), which extend the LLN under sublinear expectations in Peng (2008b) and the strong LLN for capacities in Chen (2010). It turns out that these theorems are natural extensions of the classical strong (weak) LLNs to the case where probability measures are no longer additive.
Highlights
The classical strong laws of large numbers (strong law of large numbers (LLN)) as fundamental limit theorems in probability theory play a fruitful role in the development of probability theory and its applications
In [1], Chen presented a strong law of large numbers for capacities induced by sublinear expectations with the notion of identically distributed (IID) random variables initiated by Peng
We prove three laws of large numbers under Peng’s sublinear expectations, which extend Peng’s law of large numbers under sublinear expectations in [8] and Chen’s strong law of large numbers for capacities in [1]
Summary
The classical strong (weak) laws of large numbers (strong (weak) LLN) as fundamental limit theorems in probability theory play a fruitful role in the development of probability theory and its applications. In [1], Chen presented a strong law of large numbers for capacities induced by sublinear expectations with the notion of IID random variables initiated by Peng. The following representation theorem for sublinear expectations is very useful (see Peng [8, 9] for the proof). The following is the notion of IID random variables under sublinear expectations introduced by Peng [5,6,7,8,9].
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