Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T16:49:08.094Z Has data issue: false hasContentIssue false
  • English
  • Français

Rates of Growth in a Class of Homogeneous Multidimensional Markov Chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

M. González ,
R. Martínez  and
M. Mota
M. González*
Affiliation:
Universidad de Extremadura
R. Martínez*
Affiliation:
Universidad de Extremadura
M. Mota*
Affiliation:
Universidad de Extremadura
*
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain.
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain.
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the asymptotic behaviour of homogeneous multidimensional Markov chains whose states have nonnegative integer components. We obtain growth rates for these models in a situation similar to the near-critical case for branching processes, provided that they converge to infinity with positive probability. Finally, the general theoretical results are applied to a class of controlled multitype branching process in which random control is allowed.

Keywords

Homogeneous multidimensional Markov chainrate of growthmultitype branching process

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 159 - 174
Copyright
© Applied Probability Trust 2006 

Footnotes

Supported by the Ministerio de Ciencia y Tecnología and the FEDER through the Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica, grant BFM2003-06074.

References

Chow, Y. S. and Teicher, H. (1997).” Probability Theory. Independence, Interchangeability, Martingales. ” Springer, New York.Google Scholar
Chung, K. L. (1967).” Markov Chains with Stationary Transition Probabilities, 2nd edn. ” Springer, New York.Google Scholar
Gentleman, R. and Ihaka, R. (1996). “R: a language for data analysis and graphics.” J. Comput. Graph. Statist. 5, 299–314.Google Scholar
González, M., Martínez, R. and Mota, M. (2005). “On the geometric growth in a class of homogeneous multitype Markov chains.” J. Appl. Prob. 42, 1015–1030.Google Scholar
González, M., Martínez, R. and Mota, M. (2005). “On the unlimited growth of a class of homogeneous multitype Markov chains.” Bernoulli 11, 559–570.CrossRefGoogle Scholar
González, M., Molina, M. and del Puerto, I. (2002). “On the class of controlled branching processes with random control functions.” J. Appl. Prob. 39, 804–815.Google Scholar
González, M., Molina, M. and del Puerto, I. (2003). “On the geometric growth in controlled branching processes with random control function.” J. Appl. Prob. 40, 995–1006.CrossRefGoogle Scholar
González, M., Molina, M. and del Puerto, I. (2005). “Asymptotic behaviour for the critical controlled branching process with random control function.” J. Appl. Prob. 42, 463–477.CrossRefGoogle Scholar
González, M., Molina, M. and del Puerto, I. (2005). “On {L}2-convergence of controlled branching processes with random control function.” Bernoulli 11, 37–46.Google Scholar
Keller, G., Kersting, G. and Rösler, U. (1987). “On the asymptotic behaviour of discrete time stochastic growth processes.” Ann. Prob. 15, 305–343.Google Scholar
Kersting, G. (1992). “Asymptotic {Γ}-distribution for stochastic difference equations.” Stoch. Process. Appl. 40, 15–28.Google Scholar
Klebaner, F. (1991). “Asymptotic behaviour of near-critical multitype branching processes.” J. Appl. Prob. 28, 512–519.CrossRefGoogle Scholar
Mode, C. J. (1971).” Multitype Branching Processes. ” Elsevier, New York.Google Scholar
Seneta, E. (1981).” {Nonnegative Matrices and Markov Chains}, 2nd edn. ” Springer, New York.Google Scholar
Sevast´yanov, B. A. and Zubkov, A. (1974). “Controlled branching processes.” Theory Prob. Appl. 19, 14–24.Google Scholar
Von Bahr, B. and Esseen, C. G. (1965). “Inequalities for the rth absolute moment of a sum of random variables, 1≤ r≤ 2.” Ann. Math. Statist. 36, 299–303.Google Scholar
Yanev, N. (1975). “Conditions for degeneracy of ɸ-branching processes with random ɸ.” Theory Prob. Appl. 20, 421–428.Google Scholar