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 Mat. Zametki, 2011, Volume 90, Issue 6, Pages 845–859 (Mi mz8863)

Limit Distributions for the Number of Particles in Branching Random Walks

E. Vl. Bulinskaya

M. V. Lomonosov Moscow State University

Abstract: We study branching random walks with continuous time. Particles performing a random walk on $\mathbb{Z}^{2}$, are allowed to be born and die only at the origin. It is assumed that the offspring reproduction law at the branching source is critical and the random walk outside the source is homogeneous and symmetric. Given particles at the origin, we prove a conditional limit theorem for the joint distribution of suitably normalized numbers of particles at the source and outside it as time unboundedly increases. As a consequence, we establish the asymptotic independence of such random variables.

Keywords: branching random walk, branching source, offspring reproduction law, Bellman–Harris branching process, probability generating function, transition rate matrix

DOI: https://doi.org/10.4213/mzm8863

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English version:
Mathematical Notes, 2011, 90:6, 824–837

Bibliographic databases:

UDC: 519.21
Revised: 27.12.2010

Citation: E. Vl. Bulinskaya, “Limit Distributions for the Number of Particles in Branching Random Walks”, Mat. Zametki, 90:6 (2011), 845–859; Math. Notes, 90:6 (2011), 824–837

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz8863
• https://doi.org/10.4213/mzm8863
• http://mi.mathnet.ru/eng/mz/v90/i6/p845

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Vatutin, V. A. Topchii, “Critical Bellman–Harris branching processes with long-living particles”, Proc. Steklov Inst. Math., 282 (2013), 243–272
2. E. Vl. Bulinskaya, “Subcritical catalytic branching random walk with finite or infinite variance of offspring number”, Proc. Steklov Inst. Math., 282 (2013), 62–72
3. E. V. Bulinskaya, “Local particles numbers in critical branching random walk”, J. Theor. Probab., 27:3 (2014), 878–898
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