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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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