We discuss uniform limit theorems for linear combinations of order statistics, when a family of weighting functions (or score functions) is involved. We also show an application of the results to derive asymptotic properties of coherent risk functionals.

By an adaptation of a method originally invented by G. Kersting [1] for the calculation of the limiting distribution of Markovian processes the central limit theorem (CLT) is proven. Only the case of equal variances is considered.

The contour process of a random binary tree t with n internal nodes is defined as the polygonal function constructed from the heights of the leaves of t (normalized by ). We show that, as n --> [infinity], the limiting contour process is identical in distribution to a Brownian excursion.