Statistical inference of subcritical strongly stationary Galton–Watson processes with regularly varying immigration

We describe the asymptotic behavior of conditional least squares estimator offspring mean for subcritical strongly stationary Galton–Watson processes with regularly varying immigration tail index ? ? ( 1 , 2 ) . The limit law is ratio two dependent stable random variables indices ? and 3 respectively, it has a continuously differentiable density function. use point process technique in proofs.