Criterion of unlimited growth of critical multidimensional stochastic models

TL;DR

This paper establishes a criterion for the unlimited growth of multidimensional stochastic models, providing insights into recurrence, transience, and size-dependent processes, with improvements over existing results.

Contribution

It introduces a new criterion for growth in multidimensional stochastic models and enhances understanding of multitype Galton-Watson processes with immigration.

Findings

Derived a criterion for positive probability of unlimited growth.

Revealed necessary and sufficient conditions for recurrence and transience.

Improved results on size-dependent multitype Galton-Watson processes.

Abstract

We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton-Watson with immigration processes. We also significantly improve some results on multitype size-dependent Galton-Watson processes.