A Branching Process with Immigration in Varying Environments

Bienaymé–Galton–Watson branching processes with varying offspring variance and an immigration component are studied in the critical case. The asymptotic formulas for the probability for non extinction are derived, in dependence of immigration component. A limit theorem is proved too.     

Local dependence functions for extreme value distributions

Kotz & Nadarajah (2002) introduced a measure of local dependence which is a localized version of the Pearson's correlation coefficient. In this paper we provide detailed analyses (both algebraic and numerical) of the form of the measure for the class of bivariate extreme value distributions. We consider, in particular, five families of bivariate extreme value distributions. We also discuss two applications of the new measure. In the first application we introduce an overall measure of correlation and produce evidence to suggest that it is superior than the usual Pearson's correlation coefficient. The second application introduces two new concepts for ordering of bivariate dependence.