For the stationary branching process X = A + sum_{i=1}^X B_i with index-one heavy tails and E[B]=b<1, the tail is asymptotic to 1/((1-b)(1+x)).
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The Boundary Principle of a Single Big Jump: Refined Asymptotics for Branching Processes with Immigration
For the stationary branching process X = A + sum_{i=1}^X B_i with index-one heavy tails and E[B]=b<1, the tail is asymptotic to 1/((1-b)(1+x)).