Multivariate Stability and Strong Limiting Behaviour of Intermediate Order Statistics

We investigate the almost sure stability of intermediate order statistics of a sample from a multivariate distribution. It is shown that the ordering of the multivariate sample—based on an increasing family of conditional quantile surfaces—enables us to use a univariate approach. In the univariate case, we characterise the strong limiting behaviour of weighted uniform quantile processes along an intermediate sequence satisfying suitable conditions of growth and regularity. From this characterisation a necessary and sufficient condition for the almost sure stability is deduced.