A. A. Balkema

Uniform Rates of Convergence to Extreme Value Distributions

This paper is intended as an introduction to the problem of finding reasonable rates of convergence to be presented at the Workshop on Rates, of convergence at the Vimeiro meeting.

Domains of attraction for exponential families

With the df F of the rv X we associate the natural exponential family of dfs F[lambda] wheredF[lambda](x)=e[lambda]x dF(x)/Ee[lambda]Xfor . Assume [lambda][infinity]=sup [Lambda][less-than-or-equals, slant][infinity] does not lie in [Lambda]. Let [lambda][short up arrow][lambda][infinity], then non-degenerate limit laws for the normalised distributions F[lambda](a[lambda]x+b[lambda]) are the normal and gamma distributions. Their domains of attractions are determined. Applications to saddlepoint and gamma approximations are considered.

Limit laws for exponential families

We study the asymptotic behaviour of the distribution functions FA as A increases to A, := sup A. If A00 = oc then FA 1 0 pointwise on {F 0 and bA, and in this case either G is a Gaussian distribution or G has a finite lower end-point yo = inf{ G > 0} and G(y yo) is a gamma distribution. Similarly, if Ac, is finite and does not belong to A then G is a Gaussian distribution or G has a finite upper end-point y, and 1 G(y, y) is a gamma distribution. The situation for sequences A, T;1 is entirely different: any distribution function may occur as the weak limit of a sequence FA,(anx + bn).

Stability of an M|G|1 queue with thick tails and excess capacity

The behavior of queues with excess capacity whose service times have very thick tails is studied.

Stopping times and sample path regularity

It is possible to introduce the concept of a stopping time in an algebraic way, and define for each stopping time a natural sigma-field. This yields a simple theorem about sample path regularity for processes with a complete separable metric state space.