Two necessary conditions for embeddability of a Galton–Watson branching process

Abstract

1.Introduction and statement of results. The most substantial contribution in recent years to the literature on the Galton–Watson embedding problem has been that of Karlin and McGregor (4) and (5), to which the readeris referred for a more complete bibliography of the subject. Their results and methods very much involve the analytic properties of the associated probability generating functions (p.g.f.'s), and most of the results they describe as ‘negative’ in the sense that they are sufficient conditions for non-embeddability. The two results here are similarly ‘negative’ but obtained from more direct probabilistic considerations.