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Motivated by risk measure, super-hedge pricing, and modeling uncertainty in finance, Shige Peng established the theory of sub-linear expectation. In this article, we derive two results of laws of large numbers in the framework of sub-linear expectations. One is the strong law of large numbers for the array of random variables, which satisfies non identical distributed and exponential negatively dependent under sub-linear expectation. The other is the weak law of large numbers for the array of random variables, which satisfies non identical distributed and Φ -negatively dependent under sub-linear expectation. These results include and extend some existing results. [ABSTRACT FROM AUTHOR]

Both Khinchin's law of large numbers (Khinchin's LLN) and Lindeberg's central limit theorem (Lindeberg's CLT) are fundamental results in probability theory. In this paper, we give the new proofs of these two theorems. A law of large numbers and a central limit theorem are proved for independent and non-identical distributed random variables. Indeed, these results include the Khinchin's LLN and Lindeberg's CLT. Our main tool is the viscosity solution theory of partial differential equation (PDE). [ABSTRACT FROM AUTHOR]

In this article, in the framework of sublinear expectation initiated by Peng, we derive a strong law of large numbers (SLLN) for negatively dependent and non identical distributed random variables. This result includes and extends some existing results. Furthermore, we give two examples of our result for applications. [ABSTRACT FROM AUTHOR]

Let $X_{i}, i\geq1$ be a sequence of random variables with different distributions $F_{i}, i\geq1$ . The partial sums are denoted by $S_{n}=\sum_{i=1}^{n} X_{i}$ , $n\geq1$ . This paper mainly investigates the precise large deviations of $S_{n}, n\geq1$ , for the widely orthant dependent random variables $X_{i}, i\geq1$ . Under some mild conditions, the lower and upper bounds of the precise large deviations of the partial sums $S_{n}$ , $n\geq1$ , are presented. [ABSTRACT FROM AUTHOR]