Local convergence of critical random trees and continuous-state branching processes

2015/03/03 by Xin He, He, Xin
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1503.00951

openalex publication_date 2015/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the local convergence of critical Galton-Watson trees and Levy trees under various conditionings. Assuming a very general monotonicity property on the functional of random trees, we show that random trees conditioned to have large functional values always converge locally to immortal trees. We also derive a very general ratio limit property for functionals of random trees satisfying the monotonicity property. Then we move on to study the local convergence of critical continuous-state branching processes, and prove a similar result. Finally we give a definition of continuum condensation trees, which should be the correct local limits for certain subcritical Levy trees under suitable conditionings.

Citations

• The maximal jump and local convergence of continuous-state branching processes

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