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Workshop on Branching Processes and Their Applications pp 79–94Cite as

Extension of the problem of extinction on Galton–Watson family trees

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Part of the book series: Lecture Notes in Statistics ((LNSP,volume 197))

Abstract

We review the existing and present new results on certain subtrees of the Galton-Watson family tree. For a positive integer N, define an N-ary subtree to be the tree of a deterministic N-splitting, rooted at the ancestor. Dekking (Amev. Math. Monthly 98:728–731, 1991) raised and answered the question how to compute the probability for a branching process to possess the binary splitting property, i.e., N = 2. Pakes and Dekking (J. Theor. Probab. 4:353–369, 1991) studied the general situation when N ≥ 2. Surprisingly, the case N ≥ 2 is studied so late, whereas the question for extinction of a branching process, i.e., non-existence of an infinite unary subtree (N = 1) has been studied extensively over the past 120–150 years.

Mathematics Subject Classification (2000): 60J80, 05C05

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Authors and Affiliations

  1. Department of Mathematics, University of Texas– Pan America, 1201 West University Drive, Edinburg, TX, 78539, USA

    George P. Yanev

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  1. George P. Yanev
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Correspondence to George P. Yanev .

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Editors and Affiliations

  1. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Miguel González Velasco

  2. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Inés M. Puerto

  3. Universidad de Extremadura, Avda. Virgen del Puerto, Plasencia, 10600, Spain

    Rodrigo Martínez

  4. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Manuel Molina

  5. Fac. Ciencias, Universidad de Extremadura, Avenida Elvas s/n, Badajoz, 06071, Spain

    Manuel Mota

  6. Facultad de Veterinaria, Universidad de Extradura, Avda Universidad s/n, Caceres, 10071, Spain

    Alfonso Ramos

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Yanev, G.P. (2010). Extension of the problem of extinction on Galton–Watson family trees. In: González Velasco, M., Puerto, I., Martínez, R., Molina, M., Mota, M., Ramos, A. (eds) Workshop on Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11156-3_6

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