Weak convergence of continuous-state branching processes with large immigration

TL;DR

This paper establishes functional limit theorems for continuous-state branching processes with immigration, revealing how different immigration regimes lead to various limiting processes such as subordinators and extremal processes.

Contribution

It introduces new limit theorems for CBIs with large tail immigration laws, expanding understanding of their asymptotic behaviors under different regimes.

Findings

Limiting processes include subordinators, CBIs, extremal processes, and extremal shot noise processes.

Different immigration regimes lead to distinct types of limiting processes.

Theorems apply to CBIs with finite first moments and heavy-tailed immigration laws.

Abstract

Functional limit theorems are established for continuous-state branching processes with immigration (CBIs), where the reproduction laws have finite first moments and the immigration laws exhibit large tails. Different regimes of immigration are identified, leading to limiting processes that are either subordinators, CBIs, extremal processes, or extremal shot noise processes.