A note on simple branching processes with infinite mean
- 1 June 1977
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 14 (04) , 836-842
- https://doi.org/10.1017/s0021900200105364
Abstract
We consider the Bienaymé–Galton–Watson process without and with immigration, and with offspring distribution having infinite mean. For such a process, {Zn } say, conditions are given ensuring that there exists a sequence of positive constants, {ρn }, such that {ρnU(Zn + 1)} converges almost surely to a proper non-degenerate random variable, where U is a function slowly varying at infinity, defined on [1, ∞), continuous and strictly increasing, with U(1) = 0, U(∞) = ∞. These results subsume earlier ones with U(t) = log t.Keywords
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