Local Convergence of Critical Random Trees and Continuous-State Branching Processes

Abstract

We study the local convergence of critical Galton–Watson trees and Lévy trees under various conditionings. Assuming a very general monotonicity property on the measurable functions of critical random trees, we show that random trees conditioned to have large function values always converge locally to immortal trees. We also derive a very general ratio limit property for measurable functions of critical random trees satisfying the monotonicity property. Finally we study the local convergence of critical continuous-state branching processes, and prove a similar result.