• Bernoulli
  • Volume 17, Number 1 (2011), 138-154.

Supercritical age-dependent branching Markov processes and their scaling limits

Krishna B. Athreya, Siva R. Athreya, and Srikanth K. Iyer

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This paper studies the long-time behavior of the empirical distribution of age and normalized position of an age-dependent supercritical branching Markov process. The motion of each individual during its life is a random function of its age. It is shown that the empirical distribution of the age and the normalized position of all individuals alive at time $t$ converges as $t→∞$ to a deterministic product measure.

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Bernoulli, Volume 17, Number 1 (2011), 138-154.

First available in Project Euclid: 8 February 2011

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age-dependent ancestral times branching empirical distribution supercritical


Athreya, Krishna B.; Athreya, Siva R.; Iyer, Srikanth K. Supercritical age-dependent branching Markov processes and their scaling limits. Bernoulli 17 (2011), no. 1, 138--154. doi:10.3150/10-BEJ264.

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  • [1] Athreya, K.B. and Kang, H.J. (1998). Some limit theorems for positive recurrent Markov Chains I and II. Adv. in Appl. Probab. 30 693–722.
  • [2] Athreya, K.B. and Ney, P. (2004). Branching Processes. New York: Dover.
  • [3] Biggins, J.D. and Kyprianou, A.E. (1997). Seneta–Heyde norming in the branching random walk. Ann. Probab. 25 337–360.
  • [4] Bingham, N.H. and Doney, R.A. (1975). Asymptotic properties of supercritical branching processes, II. Crump–Mode and Jirina processes. Adv. in Appl. Probab. 7 66–82.
  • [5] Durrett, R. and Liggett, T.M. (1983). Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 275–301.
  • [6] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [7] Nerman, O. and Jagers, P. (1984). The stable double infinite pedigree process of supercritical branching populations. Z. Wahrsch. Verw. Gebiete 65 445–460.
  • [8] Kallenberg, O. (2002). Foundation of Modern Probability Theory. New York: Springer.
  • [9] Samuels, M. (1971). Distribution of the branching process population among generations. J. Appl. Probab. 8 655–667.