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Recurrence criteria for multi-dimensional Markov chains and multi-dimensional linear birth and death processes

  • Harry Kesten (a1)
Abstract

Criteria are established for a discrete-time Markov process {Xn } n≧0 in R d to have strictly positive, respectively zero, probability of escaping to infinity. These criteria are mainly in terms of the mean displacement vectors μ(y) = E{X n+1|Xn = y} – y, and are essentially such that they force a deterministic process w.p.1 to move off to infinity, respectively to return to a compact set infinitely often. As an application we determine of most two-dimensional birth and death processes with rates linearly dependent on the population, whether they can escape to infinity or not.

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[1] Breiman, L. (1968) Probability. Addison-Wesley, San Francisco.
[2] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.
[3] Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco.
[4] Friedman, A. (1973) Wandering out to infinity of diffusion processes. Trans. Amer. Math. Soc. 184, 185203.
[5] Iglehart, D. L. (1964) Multivariate competition processes. Ann. Math. Statist. 35, 350361.
[6] Karlin, S. and Kaplan, N. (1973) Criteria for extinction of certain population growth processes with interacting types. Adv. Appl. Prob. 5, 183199.
[7] Kesten, H. (1970) Quadratic transformations: a model for population growth II. Adv. Appl. Prob. 2, 179228.
[8] Kesten, H. (1972) Limit theorems for stochastic growth models. Adv. Appl. Prob. 4, 193232 and 393–428.
[9] Kesten, H. and Stigum, B. P. (1975) Balanced growth under uncertainty in decomposable economics. In Essays on Economic Behaviour Under Uncertainty, ed. Balch, M. S., McFadden, D. L. and Wu, S. Y., American Elsevier, New York, 339381.
[10] Lamperti, J. (1960) Criteria for the recurrence or transience of stochastic processes I. J. Math. Anal. Appl. 1, 314330.
[11] Milch, P. R. (1968) A multi-dimensional linear growth birth and death process. Ann. Math. Statist. 39, 727754.
[12] Pinsky, M. A. (1974) Stochastic stability and the Dirichlet problem. Comm. Pure Appl. Math. 27, 311350.
[13] Reuter, G. E. H. (1961) Competition processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 421430.
[14] Tanny, D. (1975) Branching Processes in Random Environments. , Cornell University.
[15] Whittle, P. (1969) Refinements of Kolmogorov's inequality. Teor. Veroyat. Primen. 14, 315317 (Theor. Prob. Appl. 14, 310–311).
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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