Asymptotic Results for Spacings of Largest Order Statistics

We prove o- and O-type asymptotic results in probability for properly normalized spacings (differences) of the largest order statistics based on i.i.d. random sample, as the size of the sample tends to infinity. In particular, compactness properties of sequences of spacings are investigated. These results are applied to study the corresponding asymptotic properties of the ratios of upper order statistics and the number of observations near extremes. A common distribution function of the sample is assumed to satisfy mild conditions on regular variation at its right endpoint.