Power limits for central order statistics: I. Continuous limit laws

This paper lists the continuous limit distributions for central order statistics normalized by power transformations, and describes their domains of attraction. One may argue that power transformations are the natural normalizations to use if one wants to study the asymptotic behaviour of central order statistics. Power transformations preserve the origin, which may be assumed to be the quantile to which the order statistics converge. Our theory gives a nice extension of the theory developed by Smirnov more than sixty year ago. For the continuous power limits treated below the resemblance with the limit theory for extremes under linear transformations is striking