Central Limit Theorem: Characteristic Functions

This lecture presents an alternative proof of the Central Limit Theorem using characteristic functions, showing that the convergence in distribution occurs if and only if the characteristic functions of the random variables converge. The proof involves demonstrating the convergence of the characteristic function of a sum of independent and identically distributed random variables to that of a Gaussian with zero mean and unit variance. By analyzing the characteristic function of the standardized sum, the lecture illustrates how the Gaussian distribution emerges as the limit due to the properties of the first and second moments, highlighting the universality of the Gaussian distribution in the large sample limit.