Asymptotic behavior of a multi-type nearly critical Galton--Watson processes with immigration

TL;DR

This paper analyzes the long-term behavior of multi-type inhomogeneous Galton--Watson processes with immigration, deriving the limit distribution when the offspring mean matrix approaches criticality.

Contribution

It provides the first comprehensive limit distribution results for multi-type processes with immigration under near-critical conditions.

Findings

Derived the limit distribution for the process

Coordinates of the limit vector may be dependent

Applicable under general convergence conditions

Abstract

Multi-type inhomogeneous Galton--Watson process with immigration is investigated, where the offspring mean matrix slowly converges to a critical mean matrix. Under general conditions we obtain limit distribution for the process, where the coordinates of the limit vector are not necessarily independent.