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

Published online by Cambridge University Press:  24 October 2008

D. R. Grey
D. R. Grey
Affiliation:
University of Sheffield
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 2 , September 1975 , pp. 339 - 343
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Athreya, K. B. & NEY, P. E.Branching processes. (Springer-Verlag; Berlin, 1972.)CrossRefGoogle Scholar
(2)Feller, W.An introduction to probability theory and its applications, Vol. I (Third Edition). (Wiley; New York, 1968.)Google Scholar
(3)Jir˘ina, M.Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (1958), 292–313.CrossRefGoogle Scholar
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(5)Karlin, S. and McGregor, J.Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Amer. Math. Soc. 132 (1968), 137–145.CrossRefGoogle Scholar