Limit theorems for bifurcating integer-valued autoregressive processes

TL;DR

This paper investigates the long-term behavior of estimators in bifurcating integer-valued autoregressive processes, establishing their convergence and distributional properties using martingale techniques.

Contribution

It provides new asymptotic results for estimators in bifurcating integer-valued autoregressive models, including almost sure convergence and limit theorems.

Findings

Almost sure convergence of estimators

Quadratic strong law established

Central limit theorem proved

Abstract

We study the asymptotic behavior of the weighted least squares estimators of the unknown parameters of bifurcating integer-valued autoregressive processes. Under suitable assumptions on the immigration, we establish the almost sure convergence of our estimators, together with the quadratic strong law and central limit theorems. All our investigation relies on asymptotic results for vector-valued martingales.