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Markovian paths to extinction

  • Peter Jagers (a1), Fima C. Klebaner (a2) and Serik Sagitov (a1)

Abstract

Subcritical Markov branching processes {Z t } die out sooner or later, say at time T < ∞. We give results for the path to extinction {Z uT , 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of x u−1 Z uT , as Z 0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by Z T-u , u > 0, and show that as Z 0= x → ∞ this process converges to a Markov process {Y u }, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.

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Copyright

Corresponding author

Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.
∗∗ Email address: jagers@math.chalmers.se
∗∗∗ Postal address: School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia.

References

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[1] Alsmeyer, G. and Rösler, U. (2002). Asexual versus promiscuous bisexual Galton–Watson processes: the extinction probability ratio. Ann. Appl. Prob. 12, 125142.
[2] Alsmeyer, G. and Rösler, U. (2006). The Martin entrance boundary of the Galton–Watson process. Ann. Inst. H. Poincaré Prob. Statist. 42, 591606.
[3] Athreya, K. and Ney, P. (1972). Branching Processes. Springer, Berlin.
[4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
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[6] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
[7] Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, Chichester.
[8] Jagers, P., Klebaner, F. C. and Sagitov, S. (2007). On the path to extinction. Proc. Nat. Acad. Sci. USA 104, 61076111.
[9] Klebaner, F. C., Rösler, U. and Sagitov, S. (2006). Transformations of Galton–Watson processes and linear fractional reproduction. Submitted.
[10] Pakes, A. G. (1989). Asymptotic results for the extinction time of Markov branching processes allowing emigration. I. Random walk decrements. Adv. Appl. Prob. 21, 243269.
[11] Pakes, A. G. (1989). On the asymptotic behaviour of the extinction time of the simple branching process. Adv. Appl. Prob. 21, 470472.
[12] Sevast'yanov, B. A. (1971). Vetvyashchiesya Protsessy. Nauka, Moscow.

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Markovian paths to extinction

  • Peter Jagers (a1), Fima C. Klebaner (a2) and Serik Sagitov (a1)

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