Résumé : I am going to discuss weak convergence in the Skorokhod space of Galton-Watson processes with immigration (GWI), properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. It will be explained that the limits are extremal shot noise processes. Interestingly, both the behavior in mean and the survival probability (especially in the subcritical case) of the underlying Galton-Watson processes without immigration affect the asymptotics in question. The sequence of the conditional expectations of GWI given the immigration forms a very particular instance of a (divergent) perpetuity. In view of this I shall also present functional limit theorems in the Skorokhod space for general divergent perpetuities and suprema of perturbed random walks which are closely related objects.
|Dernière modification : Monday 24 January 2022||Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr|