We provide a generalization of Theorem 1 in Bartkiewicz et al. (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 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.
We would like to thank Thomas Mikosch for his suggestion to use the anti-clustering type condition (2.6) presented in Lemma 2.5, which will appear in his forthcoming book (2022+) written jointly with Olivier Wintenberger. We would like to thank the referee for the comments that helped us improve the paper. Mátyás Barczy is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Fanni K. Nedényi is supported by the UNKP-18-3 New National Excellence Program of the Ministry of Human Capacities. Gyula Pap was supported by grant NKFIH-1279-2/2020 of the Ministry for Innovation and Technology, Hungary.
"Convergence of partial sum processes to stable processes with application for aggregation of branching processes." Braz. J. Probab. Stat. 36 (2) 315 - 348, June 2022. https://doi.org/10.1214/21-BJPS528