Electronic Communications in Probability

On recurrence and transience of multivariate near-critical stochastic processes

Götz Kersting

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We obtain complementary recurrence/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].

Article information

Electron. Commun. Probab. Volume 22 (2017), paper no. 7, 12 pp.

Received: 17 May 2016
Accepted: 27 December 2016
First available in Project Euclid: 12 January 2017

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Digital Object Identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Markov property recurrence transience Lyapunov function martingale

Creative Commons Attribution 4.0 International License.


Kersting, Götz. On recurrence and transience of multivariate near-critical stochastic processes. Electron. Commun. Probab. 22 (2017), paper no. 7, 12 pp. doi:10.1214/16-ECP39. https://projecteuclid.org/euclid.ecp/1484190065

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  • [1] E. Adam (2015). Criterion for unlimited growth of critical multidimensional stochastic models. arXiv:1502.04046 [math.PR]. To appear in Adv. Appl. Probab.
  • [2] R. Durrett. Probability: Theory and examples. Cambridge University Press, 4th edition, 2010.
  • [3] M. González, R. Martínez, and M. Mota (2005). On the unlimited growth of a class of homogeneous multityp Markov chains. Bernoulli 11: 559–570.
  • [4] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1990.
  • [5] G. Kersting (1986). On recurrence and transience of growth models. J. Appl. Probab. 23: 614–625.
  • [6] G. Kersting (2016). Recurrence and transience of near-critical multivariate growth models: criteria and examples. Branching processes and their applications, ed. I. M. del Puerto et al. Lecture Notes in Statistics, 219: 207–218.
  • [7] F. Klebaner (1991). Asymptotic behavior of near-critical multitype branching processes. J. Appl. Probab. 28: 512–519.
  • [8] J. Lamperti (1960). Criteria for recurrence or transience of stochastic processes. J. Math. Anal. Appl. 1: 314–330.
  • [9] E. Seneta. Non-negative matrices and Markov chains. Springer, 2nd edition, 1981.