On Aggregation of Subcritical Galton-Watson Branching Processes with Regularly Varying Immigration

We study an iterated temporal and contemporaneous aggregation ofNindependent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index alpha is an element of (0,2). We show that limits of finite-dimensional distributions of appropriately centered and scaled aggregated partial-sum processes exist when first taking the limit asN -> infinity and then the time scalen -> infinity. The limit process is an alpha-stable process if alpha is an element of (0,1) ? (1,2) and a deterministic line with slope 1 if alpha= 1.