The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911141178900480 |
|---|---|
| author | Zhao, Yunfan |
| author_facet | Zhao, Yunfan |
| contents | We analyze the stationary tail of a fixed-point equation arising in branching processes with state-independent immigration, when both immigration and offspring distributions have heavy tails with boundary index one. We prove that \[ P(X > x) \sim \frac{1}{(1-b)(1+x)}, \quad x \to \infty, \] and provide a refined asymptotic identifying negligible logarithmic corrections. Our approach develops a closure principle for subexponential summations and a cluster expansion representation, which disentangles immigration- and branching-driven extremes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration Zhao, Yunfan Probability 60J80, 60G70, 60K25 We analyze the stationary tail of a fixed-point equation arising in branching processes with state-independent immigration, when both immigration and offspring distributions have heavy tails with boundary index one. We prove that \[ P(X > x) \sim \frac{1}{(1-b)(1+x)}, \quad x \to \infty, \] and provide a refined asymptotic identifying negligible logarithmic corrections. Our approach develops a closure principle for subexponential summations and a cluster expansion representation, which disentangles immigration- and branching-driven extremes. |
| title | The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration |
| topic | Probability 60J80, 60G70, 60K25 |
| url | https://arxiv.org/abs/2509.05650 |