Classical and Modern Branching Processes pp 257-262 | Cite as

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

Chapter

## 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.

## Key words

Galton-Watson branching tree tree-indexed equipolar## Preview

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## References

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## Copyright information

© Springer Science+Business Media New York 1997