Recurrence and transience of near-critical multivariate growth models: criteria and examples

TL;DR

This paper establishes criteria for recurrence and transience in near-critical multivariate stochastic growth models, with applications to population processes like Galton-Watson models, enhancing understanding of their long-term behavior.

Contribution

It provides new recurrence and transience criteria for multivariate stochastic processes with non-linear dynamics, including population-dependent models.

Findings

Derived criteria for recurrence and transience in multivariate models

Applied criteria to Galton-Watson processes with population dependence

Enhanced understanding of long-term behavior of near-critical stochastic processes

Abstract

We discuss complementary recurrence and transience criteria for stochastic processes $(X_n)_{n \ge 0}$ with values in the $d$-dimensional orthant $\mathbb R^d_+$ fulfilling a non-linear stochastic equation of the form $X_{n+1}=MX_n+g(X_n)+ \xi_n$ with a primitive matrix $M$ and random noise $\xi_n$ and obeying a weak Markov property. As examples we discuss bisexual Galton-Watson processes and multivariate Galton-Watson processes, which both may be population size dependent.