Comparing individual contributions in a strongly interacting system of stochastic growth processes can be a very difficult problem. This is particularly the case when new growth processes are initiated depending on the state of previous ones and the growth rates of the individual processes are themselves random. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the growth processes are jointly embedded into continuous time so that their evolutions after initiation become approximately independent while we retain some control over the initiation times. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that enables a comparison of the state of the processes initiated in a critical window and therefore allows an asymptotic description of the extremal process. In this paper we prove a versatile limit theorem of this type and show how this tool can be applied to obtain novel asymptotic results for a variety of interesting stochastic processes. These include (a) the maximal degree in different types of preferential attachment networks with fitnesses like the well-known Bianconi-Barabási tree and a network model of Dereich, (b) the most successful mutant in branching processes evolving by selection and mutation, and (c) the ratio between the largest and second largest cycles in a random permutation with random cycle weights, which can also be interpreted as a disordered version of Pitman’s Chinese restaurant process.

在随机增长过程之间相互作用强烈的系统中比较个人贡献可能是一个非常困难的问题。当根据先前状态的状态启动新的增长过程并且各个过程的增长率本身是随机的时，尤其如此。我们提出了一种解决此类问题的新颖技术，并展示了如何将其应用到产生新见解和令人惊讶结果的广泛示例中。该方法依赖于两个步骤：在第一步（高度依赖问题）中，将生长过程共同嵌入到连续的时间中，以便在启动后它们的演变变得大致独立，同时我们对启动时间保持一定控制。一旦实现这样的嵌入，第二步是应用泊松极限定理，该定理能够比较临界窗口中启动的过程的状态，因此可以渐进地描述极值过程。在本文中，我们证明了这种类型的通用极限定理，并展示了该工具如何可用于获得各种有趣的随机过程的新颖渐近结果。其中包括（a）在不同类型的具有适当适应性的优先连接网络中的最高程度，例如著名的Bianconi-Barabási树和Dereich的网络模型；（b）在分支过程中通过选择和突变演变的最成功的突变，以及（c）具有随机循环权重的随机置换中最大循环与第二大循环之间的比率，