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On recurrence and transience of multivariate near-critical stochastic processes. (English) Zbl 1357.60095
Summary: We obtain complementary recurrence/transience criteria for processes $$X=(X_n)_{n\geq 0}$$ 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 $$(\xi_{n+1})_{n\geq 0}$$ are such that $$X$$ obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [the author, J. Appl. Probab. 23, 614–625 (1986; Zbl 0611.60084)].
##### MSC:
 60J99 Markov processes 60J05 Discrete-time Markov processes on general state spaces 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60G42 Martingales with discrete parameter
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