Asymptotic Theory of Order Statistics Under Power Normalization

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. For extreme order statistics, in a recent paper by Sreehari (2009), the domains of attraction for general nonlinear normalizations are fully characterized, including the question of how to choose suitable norming functions from the knowledge of the sample distribution. The main aim of this thesis is to extend the work of Pancheva (1984) and Sreehari (2009) for the maximum order statistics to order statistics with variable ranks. It is shown that, in contrast to the case of extremes, the well known limit distributions for central order statistics under linear normalization (c.f Smirnov 1952) coincides with those under power normalization. In the case of intermediate order statistics all possible limit distributions under power normalization are derived from corresponding results in the extremal case. Furthermore, the corresponding domains of attractions of these possible limits are derived.