The use of Power Normalization as a New Trend in the Order Statistics Limit Theory

In the last two decades E. Pancheva and her collaborators were investigating various limit theorems for extremes using a wider class of normalizing mapping than the linear ones to get a wider class of limit laws. This wider class of extreme limits can be used in solving approximation problems. This review and expository paper is about this new trend in the limit theory of order statistics. We focus on the use of the power normalizing mapping. The review is given covering the possible limit laws of extreme, central and intermediate order statistics under power normalization. The paper also traces the domains of attraction of these possible limits. The final section focuses on the statistical inference about the upper tail of a distribution function by using the power normalization. Moreover, two models for generalized Pareto distribution under power normalization are given