Central Limit Theorem: Convergence in Distribution

This lecture introduces the central limit theorem, focusing on convergence in distribution. The instructor explains the concept of convergence in distribution and presents an equivalent criterion for it. The lecture covers the relationship between random variables with the same distribution and the convergence of expectations. The instructor demonstrates how to characterize convergence in distribution using continuous and bounded functions. The proof of convergence in distribution is illustrated through approximating functions and applying the monotone convergence theorem. Additionally, the lecture discusses the use of regular functions, specifically splines, to improve the convergence criterion. The equivalence between convergence in distribution and the convergence of expectations for regular functions is highlighted, emphasizing the importance of this criterion in probability theory.

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