Abstract
The skip-free property of the super-critical linear birth and death process with a linear catastrophe component is exploited to obtain an almost sure convergence theorem for the population size without any extra moment assumptions. This completes some earlier work of the author. A proof is sketched of a central limit theorem for the logarithm of the population size. Identification of the form of the norming constants for this result requires estimates of the extinction probabilities as functions of the initial population size. This is related to moment conditions on the decrement distribution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. Washington: National Bureau of Standards 1964
Barbour, A. D.: The asymptotic behaviour of birth and death and some related processes. Adv. Appl. Probab. 7, 28–43 (1975)
Brockwell, P. J.: The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Probab. 17, 42–52 (1985)
Ezhov, I. I., Reshetnyak, V. N.: A modification of the branching process. Ukranian Math. J. 35, 28–33 (1983)
Pakes, A. G.: Some limit theorems for Markov chains with applications to branching processes. In: Williams, E. J. (ed.). Studies in probability and statistics, pp. 21–39. Jerusalem: Jerusalem Academic Press 1974
Pakes, A. G.: Limit theorems for the population size of a birth and death process allowing catastrophes. J. Math. Biol. 25, 307–325 (1987)
Pakes, A. G.: Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements. Adv. Appl. Probab., to appear
Pakes, A. G., Pollett, P. K.: The supercritical birth, death and catastrophe process: Limit theorems on the set of extinction. Stochastic Proc. Appl., submitted
Author information
Authors and Affiliations
Additional information
Most of this research was carried out at Colorado State University with the partial support of N.S.F. grant DMS-8501763.
Rights and permissions
About this article
Cite this article
Pakes, A.G. The supercritical birth, death and catastrophe process: Limit theorems on the set of non-extinction. J. Math. Biology 26, 405–420 (1988). https://doi.org/10.1007/BF00276370
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276370