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Local limit theorems for non-critical Galton–Watson processes by or without immigration

Published online by Cambridge University Press:  14 July 2016

R. Höpfner
R. Höpfner*
Affiliation:
Johannes-Gutenberg-Universität Mainz
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Abstract

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn (i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.

Keywords

GALTON-WATSON PROCESSESGALTON-WATSON PROCESSES WITH IMMIGRATIONNON-CRITICAL CASELOCAL LIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 2 , June 1982 , pp. 262 - 271
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Feller, W. (1971) An Introduction to Probability Theory and its Applictions, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[3] Lamperti, J. (1967) Limiting distributions for branching processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2 (2), 225–241.Google Scholar
[4] Mellein, B. (1979) Der kritische Galton-Watson-Prozeß mit Immigration — lokale Grenz-wertsätze und approximierende Diffusionen. Dissertation, Mainz.Google Scholar