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

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