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

TL;DR

This paper investigates the asymptotic distribution of the conditional least squares estimator for the offspring mean in subcritical, strongly stationary Galton--Watson processes with heavy-tailed immigration, revealing a complex limit law involving dependent stable variables.

Contribution

It provides a detailed asymptotic analysis of the estimator's distribution in processes with regularly varying immigration, a novel contribution in this context.

Findings

Limit law is a ratio of dependent stable variables with indices α/2 and 2α/3.

The limit distribution has a continuously differentiable density.

The proof employs point process techniques.

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.