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.