More info needed on second order regular variation in extreme value theory My question is very general. I am learning extreme value theory to examine tail behavior. The concept of regular variation is still too vague to me. Could anyone help to provide more info to clarify? Any thoughts on its importance on probability theory?
 A: The concepts of slowly varying, regular varying and second order regular varying functions are used in extreme value statistics to provide regularity conditions on the behavior of the tail of a distribution function to be able to prove theorems. They can be thought of as smoothness conditions for the tail at infinity. 
The concepts are crucial to extreme value statistics, where you need some assumptions about the tail of the distribution function. One could just assume that the distribution function was from a parametric class, a Frechet distribution, for example, but this distribution may not fit the data, and one may only be interested in the tail of the distribution, in which case one would not want to make assumptions about the entire distribution function. 
Regular variation is used to give a semi-parametric class of distribution functions where we, for instance, know how the extremes behave according to the Fisher-Tippett-Gnedenko Theorem which classifies distributions into what is called the domains of attraction for one of the three extreme value distributions. 
The tail index can then be estimated, the classical estimator being the Hill estimator. More regularity, like second order regular variation, comes into the picture when we want to prove distributional results about the estimators. These conditions are technical are somewhat difficult to comprehend from an intuitive point of view, but they assure that the tail of the distribution function is "sufficiently nice". 
I good book I can recommend is Extreme Value Theory. An Introduction by Laurens de Haan and Ana Ferreira. 
