Asymptotic Distribution of the Maximum Interpoint Distance in a Sample of Random Vectors with a Spherically Symmetric Distribution

Extreme Value theory is part and parcel of any study of order statistics in one-dimension. Our aim here is to consider such large sample theory for the maximum distance to the origin, and the related maximum “interpoint distance,” in multi-dimensions. We show that for the spherically symmetric families of densities, these statistics have a Gumbel type limit, generalizing several existing results. We also discuss the other two types of limit laws and suggest some still open problems. This work complements our earlier study on the minimum interpoint distance.