A time series approach to the study of the simple subcritical Galton–Watson process with immigration

Abstract

The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).