Proof and Application of Law of Large Number and Central Limit Theorem

Abstract

The law of large number (LLN) and central limit theorem (CLT) have been important in probability, statistics, finance, and other fields for several centuries. They are also core theorems in probability. Through the work of Laplace, Chebyshev, Markov, Lyapunov, Bernoulli, and other famous mathematicians, the proof and forms of LLN and CLT have been greatly improved and expanded. They enable people to extract useful information from a large amount of data for statistical inference and prediction. Nevertheless, more scholars nowadays propose slightly improved proof for them, which may inspire people to use them better and understand their core ideas inside their elegant form. This study unravels certain proofs and applications of these two theorems to try to inspire scholars some further study of these theorems. This article uses a combination of proof of LLN and CLT and their application in different fields. Using different lemmas, such as Markov’s inequality and Chebyshev’s inequality, it discusses the process of different proofs of LLN and CLT. Based on many scholars’ research, it summarizes the application of LLN and CLT in many different fields. This article can provide learners a fundamental idea of LLN and CTL. These can help readers quickly catch the important points of LLN and CLT, which can help many different investigations in different fields, including Math, Physics, Economics, and various fields with data analysis.