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Another look at Second order condition in Extreme Value Theory

This note compares two approaches both alternatively used when establishing normality theorems in univariate Extreme Value Theory. When the underlying distribution function ($df$) is the extremal domain of attraction, it is possible to use representations for the quantile function and regularity conditions (RC), based on these representations, under which strong and weak convergence are valid. It is also possible to use the now fashion second order condition (SOC), whenever it holds, to do the same. Some authors usually favor the first approach (the SOC one) while others are fond of the second approach that we denote as the representational one. This note aims at comparing the two approaches and show how to get from one to the other. The auxiliary functions used in each approach are computed and compared. Statistical applications using simultaneously both approaches are provided. A final comparison is provided.