On recurrence and transience of multivariate near-critical stochastic processes

TL;DR

This paper establishes criteria for recurrence and transience of multivariate stochastic processes near criticality, extending known one-dimensional results to higher dimensions with non-linear dynamics and weak Markov properties.

Contribution

It provides new recurrence and transience criteria for multivariate processes with non-linear updates and weak Markov properties, generalizing previous 1D results.

Findings

Criteria for recurrence and transience in multivariate processes.

Extension of 1D criteria to higher dimensions.

Applicable to processes with non-linear dynamics and weak Markov properties.

Abstract

We obtain complementary recurrence and transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi_{n+1}$. Here $M$ denotes a primitive matrix having Perron-Frobenius eigenvalue 1, and $g$ denotes some function. The conditional expectation and variance of the noise $(\xi_{n+1})_{n \ge 0}$ are such that $X$ obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [5].