Gaussian Random Vectors

This lecture delves into the definition of Gaussian random vectors, starting with the classical definition of a Gaussian random variable. The instructor extends this definition to Gaussian random vectors, emphasizing that any linear combination of components should also be Gaussian. The lecture explores the concept of independence in Gaussian random vectors, showcasing a proposition that states the independence of Gaussian random variables with zero covariance. Additionally, the instructor highlights a counterintuitive fact where two Gaussian random variables may not form a Gaussian random vector when combined. The lecture concludes with a discussion on the characterization of Gaussian random vectors, emphasizing the relationship between covariance and independence in Gaussian random vectors.