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

Abstract

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