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

Abstract. We study an iterated temporal and contemporaneous aggregation of N independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index α ∈ (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 as N → ∞ and then the time scale n→ ∞ . The limit process is an α -stable process if α ∈ (0 , 1) ∪ ( 1 , 2 ) and a deterministic line with slope 1 if α = 1.