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

Xue Meng; Miaomiao Gao; Feng Hu

doi:10.3934/mfc.2022017

Article Contents

2023, Volume 6, Issue 2: 190-202. Doi: 10.3934/mfc.2022017

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

1.
School of Statistic and Data Science, Qufu Normal University, Qufu 273165, Shandong, China
2.
Department of Mathematics, Jining University, Qufu 273199, Shandong, China

^*Corresponding author: Feng Hu
^*Corresponding author: Feng Hu

Received: September 2021

Revised: February 2022

Early access: June 2022

Published: May 2023

The third author is supported by National Science Foundation of China (No.11801307) and Natural Science Foundation of Shandong Province (No. ZR2021MA009)

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Abstract

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).

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Mathematics Subject Classification: Primary: 60H10; Secondary: 35D40.

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