Note on Strong Law of Large Number under Sub-linear Expectation
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The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental the- ories in probability and statistics. These limit theorems are proved under additivity of probabilities and expectations. In this paper, we investigate strong law of large numbers under sub-linear expectation which generalize the classical ones. We give strong law of large numbers under sub-linear expectation with respect to the partial sums and some conditions sim- ilar to Petrov’s. It is an extension of the classical Chung type strong law of large numbers of Jardas et al.’s result. As an application, we ob- tain Chung’s strong law of large number and Marcinkiewicz’s strong law of large number for independent and identically distributed random vari- ables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.
Abstract
The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental the- ories in probability and statistics. These limit theorems are proved under additivity of probabilities and expectations. In this paper, we investigate strong law of large numbers under sub-linear expectation which generalize the classical ones. We give strong law of large numbers under sub-linear expectation with respect to the partial sums and some conditions sim- ilar to Petrov’s. It is an extension of the classical Chung type strong law of large numbers of Jardas et al.’s result. As an application, we ob- tain Chung’s strong law of large number and Marcinkiewicz’s strong law of large number for independent and identically distributed random vari- ables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.
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