Extremal limit laws for a class of bivariate Poisson vectors

Abstract

It is well known that conventional extreme value limit laws break down for the Poisson distribution: no normalization can be found to avoid degeneracy of the limit law of sample maxima. Anderson et al. (Ann. Appl. Probab. 7 (1997) 953) tackled this problem with a triangular array argument, letting both the sample size and Poisson mean grow at appropriate rates. This leads to a Gumbel limit law for sample maxima. In applications, this means that it may be appropriate to model extremes of Poisson processes using standard extreme value models and techniques. This paper extends the limit results to a class of bivariate Poisson distributions. Suitably normalized, and with a degree of dependence that is also permitted to grow at a suitable rate, we find that the limit distribution corresponds to the class of bivariate extreme value models that would have arisen, had the population been bivariate normal, cf. Hüsler and Reiss (Statist. Probab. Lett. 7 (1989) 283). This adds weight to the argument that, for practical applications involving Poisson variables, even in the presence of dependence, standard extreme value models can be applied, despite the degeneracy that arises by applying the usual asymptotic argument.