Geodesic
Weakly supercritical branching process in unfavourable environment
Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 3-19
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Let $\{Z_{n}\}$ be a weakly supercritical branching process in a random environment, and $\{S_{n}\}$ be its associated random walk. We consider a natural martingale $W_{n}=Z_{n}\exp(-S_{n})$, where $n\geq 0$. We prove two limit theorems for the random process $W_{\lfloor nt\rfloor}$, where $t\in [0,1]$, which is considered either under the condition on the unfavourable environment $\{\max_{1\leq i\leq n}S_{i}\}$ or under the condition on the unfavourable environment $\{S_{n}\leq u\}$, where $u$ is some positive constant.
Keywords: weakly supercritical branching process in a random environment, conditional functional limit theorems. 
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V. I. Afanasyev. Weakly supercritical branching process in unfavourable environment. Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 3-19. http://geodesic.mathdoc.fr/item/DM_2022_34_3_a0/

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