## The survival probability of a critical multi-type branching process in i.i.d. random environment.(English)Zbl 1445.60063

The authors study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments. Under suitable assumptions, including that the random environments admit so-called linear-fractional multidimensional generating functions, the authors obtain an optimal result, it is shown that the survival probability up to time $$n$$ multiplied by $$\sqrt{n}$$ converges to some positive limit as $$n\to\infty$$. It is explained that the main difficulty which appears in the proofs is that the fluctuations of products of some random matrices have to be investigated, and there were no available results in this direction at that time. In case of general generating functions of the random environments, the authors also prove a weaker result.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K37 Processes in random environments 60F17 Functional limit theorems; invariance principles
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### References:

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