Sharpness of second moment criteria for branching and tree-indexed processes

Robin Pemantle

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arXiv · math/0404090

Sharpness of second moment criteria for branching and tree-indexed processes

Robin Pemantle

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Abstract

A class of branching processes in varying environments is exhibited which become extinct almost surely even though the means M_n 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.

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Robin Pemantle. 2004-04-05. Sharpness of second moment criteria for branching and tree-indexed processes. https://arxiv.org/abs/math/0404090

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