Skip to main content Accessibility help

Asymptotic behaviour of Markov population processes by asymptotically linear rate of change

  • F. C. Klebaner (a1)


Multidimensional Markov processes in continuous time with asymptotically linear mean change per unit of time are studied as randomly perturbed linear differential equations. Conditions for exponential and polynomial growth rates with stable type distribution are given. From these conditions results on branching models of populations with stabilizing reproduction for near-supercritical and near-critical cases follow.


Corresponding author

Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, VIC 3052, Australia.


Hide All

Research supported by the Australian Research Council grant A68930440.



Hide All
Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.
Beckenbach, E. F. and Bellman, R. (1965) Inequalities. Springer-Verlag, New York.
Cohn, H. and Klebaner, F. C. (1986) Geometric rate of growth in Markov chains with applications to population size dependent models with dependent offspring. Stoch. Anal. Appl. 4, 283307.
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes. Wiley, New York.
Hamza, K. and Klebaner, F. C. (1993) Conditions for regularity and integrability of Markov chains. J. Appl. Prob. To appear.
Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.
Joffe, A. and Metivier, M. (1986) Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18, 2065.
Keller, G., Kersting, G. and Rosler, U. (1987) On the asymptotic behaviour of discrete time stochastic growth processes. Ann. Prob. 15, 305343.
Kersting, G. (1986) On recurrence and transience of growth models. J. Appl. Prob. 23, 614625.
Kersting, G. (1990) Some properties of stochastic difference equations. In Stochastic Modelling in Biology, ed. Tautu, P., pp. 328339. World Scientific, Singapore.
Kersting, G. (1991) A Law of Large Numbers for stochastic difference equations. Stoch. Proc. Appl. 40, 114.
Kersting, G. (1991) Asymptotic Gamma distribution for stochastic difference equations. Stoch. Proc. Appl. 40, 1528.
Kesten, H. (1971) Some non-linear stochastic growth models. Bull. Amer. Math. Soc. 77, 492511.
Kesten, H. (1972) Limit theorems for stochastic growth models. Adv. Appl. Prob. 4, 193232, 393-428.
Klebaner, F. C. (1989) Geometric growth in near-supercritical population size dependent multitype Galton-Watson processes. Ann. Prob. 17, 14661477.
Klebaner, F. C. (1989) Linear growth in near-critical population size dependent multitype Galton-Watson processes. J. Appl. Prob. 26, 431445.
Klebaner, F. C. (1991) Asymptotic behaviour of near-critical multitype branching processes. J. Appl. Prob. 28, 512519.
Kurtz, T. (1978) Diffusion approximations for branching processes. In Branching Processes (Advances in Probability 5), ed. Joffe, A. and Ney, P., pp. 262292. Marcel Dekker, New York.
Kuster, P. (1983) Generalized Markov branching processes with state-dependent offspring distributions. Z. Wahrscheinlichkeitsth. 64, 475503.
Kuster, P. (1984) Cooperative birth processes with linear or sublinear intensity. Stoch. Proc. Appl. 10, 312324.
Protter, P. (1990) Stochastic Integration and Differential Equations. Springer-Verlag, Berlin.
Reinhard, I. (1990) The qualitative behaviour of some slowly growing population-dependent Markov branching processes. In Stochastic Modelling in Biology, ed. Tautu, P., pp. 267277, World Scientific, Singapore.
Rittgen, W. (1980) Positive recurrence of multidimensional population-dependent branching processes. In Biological Growth and Spread. Lecture Notes in Biomathematics 38, pp. 98108, Springer-Verlag, Berlin.
Rittgen, W. (1987) The qualitative behaviour of population-dependent Markov branching processes I and II, SFB 123. Preprint No. 423 and 424, University of Heidelberg.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed