Convergence of partial sum processes to stable processes with application for aggregation of branching processes

Abstract

We provide a generalization of Theorem 1 in Bartkiewicz, Jakubowski, Mikosch and Wintenberger (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton--Watson branching process with regularly varying immigration having index in $(0,1)\cup (1,4/3)$ in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

46 pages. In Appendices we recall some notions and statements from arXiv:1906.00373