Stability Relationships in Distributions

This lecture discusses the importance of distributions satisfying stability relationships, such as the max stability condition. The instructor explains how the Generalized Extreme Value (GEV) and Generalized Pareto distributions are used for modeling, particularly in extrapolating into the tail of the distribution. The lecture covers the concept of domains of attraction, which define the collections of distributions leading to specific distribution limits for maxima. The instructor also delves into the penultimate approximation, a method to improve distribution approximations for finite samples. The limitations of extreme value distributions are explored, highlighting cases where no limiting distribution exists. The lecture concludes with a detailed proof of Lemma 4, showcasing the application of mathematical theorems in statistical analysis.