Recurrence and Transience of Near-Critical Multivariate Growth Models: Criteria and Examples

Abstract

We consider stochastic processes (X _n ) _n ≥ 0 taking values in $\mathbb{R}_{+}^{d} =\{ (x_{1},\ldots,x_{d})^{T} \in \mathbb{R}^{d}: x_{i} \geq 0\}$, i.e. the d-dimensional orthant, and adapted to some filtration $(\mathcal{F}_{n})_{n\geq 0}$, which satisfy an equation of the form

$$\displaystyle{X_{n+1} = MX_{n} + g(X_{n}) +\xi _{n}\,\quad n \in \mathbb{N}_{0}\,}$$

with a d × d matrix M having non-negative entries, with a function $g: \mathbb{R}_{+}^{d} \rightarrow \mathbb{R}^{d}$, and with random fluctuations ξ _n  = (ξ _n1, …, ξ _nd )^T satisfying

$$\displaystyle{\mathbf{E}[\xi _{n}\mid \mathcal{F}_{n}] = 0\ \text{ a.s.}}$$