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) 1 (1-b) (1+x), x, \ 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.
Yuhong Zhao (Sat,) studied this question.