A Short and Elementary Proof of the Central Limit Theorem by Individual Swapping

Abstract

We present a short proof of the central limit theorem which is elementary in the sense that no knowledge of characteristic functions, linear operators, or other advanced results are needed. Our proof is based on Lindeberg’s trick of swapping a term for a normal random variable in turn. The modifications needed to prove the stronger Lindeberg–Feller central limit theorem are addressed at the end.

MSC:

• Primary 60F05

Acknowledgment

This work is supported by the National Research Foundation of Korea grants NRF-2019R1A5A1028324 and NRF-2017R1A2B2001952. The author would like to thank the reviewers for valuable suggestions. He would also like to thank his friend Kihyun Kim for the comments he gave on the first version of this manuscript.

Additional information

Funding

This work is supported by the National Research Foundation of Korea grants NRF-2019R1A5A1028324 and NRF-2017R1A2B2001952. The author would like to thank the reviewers for valuable suggestions. He would also like to thank his friend Kihyun Kim for the comments he gave on the first version of this manuscript.