From probability to convergence: exploring the foundations of the central limit theorem

Abstract

[en] This thesis embarks on a mathematical journey, delving into the depths of probability theory and a comprehensive exploration of the Central Limit Theorem (CLT). The study begins with fundamental concepts such as characteristic functions and basic probability theory, continuing with the laws of large numbers. Afterwards, the narrative progresses to focus on various versions of the CLT. Essential theorems, including De Moivre-Laplace’s, Lindeberg-Lévy’s, and Lyapunov’s, are studied, offering insights into the universal significance of the CLT. The journey concludes with a glimpse of the constraints of this impactful theorem, including the convergence of the compound Poisson distribution. This work contributes to a nuanced understanding of probability theory as well as serves as a guide through the elegance and applicability of the CLT.