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

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.