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Branching processes in nearly degenerate varying environment
- Part of
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- Journal:
- Journal of Applied Probability / Volume 61 / Issue 4 / December 2024
- Published online by Cambridge University Press:
- 10 May 2024, pp. 1107-1126
- Print publication:
- December 2024
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- Article
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We investigate branching processes in varying environment, for which
$\overline{f}_n \to 1$ and 
$\sum_{n=1}^\infty (1-\overline{f}_n)_+ = \infty$, 
$\sum_{n=1}^\infty (\overline{f}_n - 1)_+ < \infty$, where 
$\overline{f}_n$ stands for the offspring mean in generation n. Since subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial limit we consider two scenarios: conditioning on nonextinction, and adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).
