Central Limit Theorem: Lindeberg's Principle

This lecture delves into the Central Limit Theorem (CLT) as a refinement of the law of large numbers, focusing on the behavior of empirical averages of random variables as n grows. The instructor explains the convergence in distribution, introduces the theorem with simple computations, and discusses the stabilization of the distribution towards a Gaussian distribution. The lecture emphasizes the significance of the Gaussian distribution in probability, showcasing how fluctuations tend towards Gaussian distribution in various setups. The instructor introduces Lindberg's principle as a method to prove the CLT, illustrating the idea of replacing random variables with Gaussian distributions to show convergence. The lecture highlights the universal nature of the CLT and its relevance in understanding random phenomena with independent contributions.