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arXiv · 2509.05650

The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration

Yunfan Zhao
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Abstract

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.

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Yunfan Zhao. 2025-09-06. The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration. https://arxiv.org/abs/2509.05650

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