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Functional normalizations for the branching process with infinite mean
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- Journal:
- Journal of Applied Probability / Volume 16 / Issue 3 / September 1979
- Published online by Cambridge University Press:
- 14 July 2016, pp. 513-525
- Print publication:
- September 1979
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- Article
- Export citation
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It is well known that, in a Bienaymé-Galton–Watson process (Zn) with 1 < m = EZ1 < ∞ and EZ1 log Z1 <∞, the sequence of random variables Znm –n converges a.s. to a non–degenerate limit. When m =∞, an analogous result holds: for any 0< α < 1, it is possible to find functions U such that α n U (Zn) converges a.s. to a non-degenerate limit. In this paper, some sufficient conditions, expressed in terms of the probability generating function of Z1 and of its distribution function, are given under which a particular pair (α, U) is appropriate for (Zn). The most stringent set of conditions reduces, when U (x)
x, to the requirements EZ1 = 1/α, EZ1 log Z1 <∞.
