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Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 3, Pages 461–484 (Mi tvp203)  

This article is cited in 18 scientific papers (total in 18 papers)

Limit theorem for critical catalytic branching random walks

V. A. Vatutina, V. A. Topchiib

a Steklov Mathematical Institute, Russian Academy of Sciences
b Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Science

Abstract: A continuous time branching random walk on the lattice $Z$ is considered in which individuals may produce children at the origin only. Assuming that the underlying Markov random walk is homogeneous and symmetric and the offspring reproduction law is critical, we describe the asymptotic behavior as $t\to\infty$ of the conditional distribution of the two-dimensional vector $(\zeta(t), \mu (t))$ (scaled in an appropriate way), where $\zeta (t)$ and $\mu(t)$ are the numbers of individuals at the origin and outside the origin at moment $t$ given $\zeta(t)>0$.

Keywords: critical Bellman–Harris branching process with two types of individuals, inhomogeneous branching random walk on the lattice of real line, limit theorems.

DOI: https://doi.org/10.4213/tvp203

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English version:
Theory of Probability and its Applications, 2005, 49:3, 498–518

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Received: 19.01.2004

Citation: V. A. Vatutin, V. A. Topchii, “Limit theorem for critical catalytic branching random walks”, Teor. Veroyatnost. i Primenen., 49:3 (2004), 461–484; Theory Probab. Appl., 49:3 (2005), 498–518

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vatutin V., Xiong J., “Some limit theorems for a particle system of single point catalytic branching random walks”, Acta Mathematica Sinica–English Series, 23:6 (2007), 997–1012  crossref  mathscinet  zmath  isi  scopus
    2. E. B. Yarovaya, “Critical branching random walks on low-dimensional lattices”, Discrete Math. Appl., 19:2 (2009), 191–214  mathnet  crossref  crossref  mathscinet  elib
    3. E. B. Yarovaya, “Criterions of the exponential growth of particles for some models of branching random walks”, Theory Probab. Appl., 55:4 (2011), 661–682  mathnet  crossref  crossref  mathscinet  isi
    4. E. V. Bulinskaya, “Catalytic branching random walk on three-dimensional lattice”, Theory Stoch. Process., 16(32):2 (2010), 23–32  mathnet  mathscinet  zmath
    5. V. A. Vatutin, V. A. Topchii, Yu. Hu, “Branching random walk in $\mathbf Z^4$ with branching at the origin only”, Theory Probab. Appl., 56:2 (2011), 193–212  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. V. A. Vatutin, V. A. Topchiǐ, “Catalytic branching random walks in $\mathbb Z^d$ with branching at the origin”, Siberian Adv. Math., 23:2 (2013), 125–153  mathnet  crossref  mathscinet  elib
    7. E. Vl. Bulinskaya, “Limit Distributions for the Number of Particles in Branching Random Walks”, Math. Notes, 90:6 (2011), 824–837  mathnet  crossref  crossref  mathscinet  isi
    8. Bulinskaya E.V., “Limit Distributions Arising in Branching Random Walks on Integer Lattices”, Lith Math J, 51:3 (2011), 310–321  crossref  mathscinet  zmath  isi  elib  scopus
    9. Yarovaya E.B., “Supercritical Branching Random Walks with a Single Source”, Comm Statist Theory Methods, 40:16 (2011), 2926–2945  crossref  mathscinet  zmath  isi  elib  scopus
    10. E. B. Yarovaya, “Spectral Properties of Evolutionary Operators in Branching Random Walk Models”, Math. Notes, 92:1 (2012), 115–131  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. V. A. Vatutin, V. A. Topchii, “Critical Bellman–Harris branching processes with long-living particles”, Proc. Steklov Inst. Math., 282 (2013), 243–272  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. Yarovaya E.B., “Branching Random Walks with Several Sources”, Math. Popul. Stud., 20:1 (2013), 14–26  crossref  mathscinet  isi  elib  scopus
    13. Yarovaya E.B., “Operators Satisfying the Schur Condition and their Applications to the Branching Random Walks”, Commun. Stat.-Theory Methods, 43:7, SI (2014), 1523–1532  crossref  mathscinet  zmath  isi  elib  scopus
    14. Carmona Ph., Hu Yu., “The Spread of a Catalytic Branching Random Walk”, Ann. Inst. Henri Poincare-Probab. Stat., 50:2 (2014), 327–351  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Yarovaya E.B., “the Structure of the Positive Discrete Spectrum of the Evolution Operator Arising in Branching Random Walks”, Dokl. Math., 92:1 (2015), 507–510  mathnet  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    16. Antonenko E., Yarovaya E., “On the Number of Positive Eigenvalues of the Evolutionary Operator of Branching Random Walk”, Branching Processes and Their Applications, Lecture Notes in Statistics, 219, eds. DelPuerto I., Gonzalez M., Gutierrez C., Martinez R., Minuesa C., Molina M., Mota M., Ramos A., Springer, 2016, 41–55  crossref  mathscinet  zmath  isi  scopus
    17. E. B. Yarovaya, “Spectral asymptotics of supercritical branching random process”, Theory Probab. Appl., 62:3 (2018), 413–431  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    18. M. V. Platonova, K. S. Ryadovkin, “O dispersii chislennosti chastits nadkriticheskogo vetvyaschegosya sluchainogo bluzhdaniya na periodicheskikh grafakh”, Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, POMI, SPb., 2019, 233–253  mathnet
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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