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

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.