A Fluctuation Limit Theorem of Branching Processes with Immigration and Statistical Applications

TL;DR

This paper establishes a fluctuation limit theorem for branching processes with immigration, showing the limit as an OU type process driven by Levy noise, with applications to statistical estimation of offspring parameters.

Contribution

It introduces a general fluctuation limit theorem for Galton-Watson processes with immigration, linking them to OU type processes driven by Levy processes.

Findings

Limit theorem for Galton-Watson processes with immigration

Asymptotic estimates for offspring mean and variance estimators

Connection to OU type processes driven by Levy noise

Abstract

We prove a general fluctuation limit theorem for Galton-Watson branching processes with immigration. The limit is a time-inhomogeneous OU type process driven by a spectrally positive Levy process. As applications of this result, we obtain some asymptotic estimates for the conditional least-squares estimator of the offspring means and variances of the offspring and immigration distributions.