New proofs of Khinchin's law of large numbers and Lindeberg's central limit theorem –PDE's approach

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).