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Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

  • Fima C. Klebaner (a1)

Abstract

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

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Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.

References

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[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.
[2] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.
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[4] Joffe, A. and Spitzer, F. (1967) On multitype branching processes with ρ ≦ 1. J. Math. Anal. Appl. 19, 409430.
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[6] Kersting, G. (1986) On recurrence and transience of growth models. J. Appl. Prob. 23, 614625.
[7] Klebaner, F. C. (1984) On population-size-dependent branching processes. Adv. Appl. Prob. 16, 3055.
[8] Mullikin, T. W. (1963) Limiting distributions for critical multitype branching processes with discrete time. Trans. Amer. Math. Soc. 106, 469494.
[9] Seneta, E. (1981) Nonnegative Matrices and Markov Chains. Springer-Verlag, New York.

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Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

  • Fima C. Klebaner (a1)

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