We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event \(\mathcal {A}_{i}(n)\) that all individuals alive at time n are offsprings of the immigrant which joined the population at time i. We study the asymptotic probability of this extreme event when n is large and i follows different asymptotics which may be related to n (i fixed, close to n, or going to infinity but far from n). In order to do so, we establish some limit theorems for random walks conditioned to stay positive or nonnegative, which are of independent interest.

中文翻译：

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随机移民环境中的关键分支过程：一个家庭的生存

我们考虑了一个iid随机环境中的一个重要分支过程，在该环境中，每个世代都有一个移民到达。我们对\（\ mathcal {A} _ {i}（n）\）事件感兴趣，该事件是所有在时间n活着的个体都是在时间i加入人口的移民的后代。我们研究当n很大并且i遵循与n有关的不同渐近性时该极端事件的渐近概率（i固定，接近n，或者趋于无穷大，但远离n）。为了做到这一点，我们建立了一些随机游动的极限定理，条件是保持正负或负非负，这是独立的。