  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.
