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Sharpness of Second Moment Criteria for Branching and Tree-Indexed Processes

  • Robin Pemantle
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

A class of branching processes in varying environments is exhibited which become extinct almost surely even though the means Mn grow fast enough so that \( \sum {M_n^{ - 1}} \) is finite. In fact, such a process is constructed for every offspring distribution of infinite variance, and this establishes the converse of a previously known fact: that if a distribution has finite variance then \( \sum {M_n^{ - 1} = \infty } \) is equivalent to almost sure extinction. This has as an immediate consequence the converse to a theorem on equipolarity of Galton-Watson trees.

Key words

Galton-Watson branching tree tree-indexed equipolar 

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References

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Robin Pemantle
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadison

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