Scaling Limits for Crump-Mode-Jagers Processes with Immigration via Stochastic Volterra Equations

2018/09/16 by Wei Xu, Xu, Wei
Economics, Econometrics and Finance · Mathematics · #45D05 #60B10 #60G57 #FOS: Mathematics #Point processes and geometric inequalities #Primary 60J80 #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #secondary 60H20

paper · pdf · doi:10.48550/arxiv.1809.05931

openalex publication_date 2018/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we firstly give a reconstruction for Crump-Mode-Jagers processes with immigration as solutions to a class of stochastic Volterra integral equations, which offers us a new insight for the evolution dynamics of age-dependent population. Based on this new representation, we prove the weak convergence of rescaled Crump-Mode-Jagers processes with immigration to a class of continuous-state branching processes with immigration. Moreover, the limits reveal that the individual law mainly changes the branching mechanism and immigration mechanism proportionally. This covers the results obtained by Lambert et al. [35] for subcritical binary Crump-Mode-Jagers processes.

Citations

• The contour of splitting trees is a Lévy process
• Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges

Related

• Functional law of large numbers and central limit theorem for Crump-Mode-Jagers branching processes
• Limit theorems for supercritical age-dependent branching processes with neutral immigration
• Non-linear branching processes and Crump-Mode-Jagers processes with interaction
• Weak convergence of continuous-state branching processes with large immigration
• Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processes