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August 1996 On the asymptotic patterns of supercritical branching processes in varying environments
Harry Cohn
Ann. Appl. Probab. 6(3): 896-902 (August 1996). DOI: 10.1214/aoap/1034968232


Let ${Z_n}$ be a branching process whose offspring distributions vary with n. It is shown that the sequence ${\max_{i>0} P(Z_n = i)}$ has a limit. Denote this limit by M. It turns out that M is positive only if the offspring variables rapidly approach constants. Let ${c_n}$ be a sequence of constants and $W_n = Z_n / c_n$. It will be proven that $M = 0$ is necessary and sufficient for the limit distribution functions of all convergent ${W_n}$ to be continuous on $(0, \infty)$. If $M > 0$ there is, up to an equivalence, only one sequence ${c_n}$ such that ${W_n}$ has a limit distribution with jump points in $(0, \infty)$. Necessary and sufficient conditions for continuity of limit distributions are derived in terms of the offspring distributions of ${Z_n}$.


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Harry Cohn. "On the asymptotic patterns of supercritical branching processes in varying environments." Ann. Appl. Probab. 6 (3) 896 - 902, August 1996.


Published: August 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0869.60075
MathSciNet: MR1410120
Digital Object Identifier: 10.1214/aoap/1034968232

Primary: 60J80
Secondary: 60F25

Keywords: branching , Galton-Watson , limit distribution , martingale , supercritical , varying environments

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 3 • August 1996
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