We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the consideration of particular models and highlight the essentials. Besides giving a unified presentation of the theory, this work investigates facets so far unexplored in previous studies. Specifically, we show how the study of the class of random allocation models and zero-range processes can serve as a backdrop for the study of the two other classes of processes central to the present work—tied-down and free renewal processes. We then present new insights on the extreme value statistics of these three classes of processes which allow a deeper understanding of the mechanism of condensation and the quantitative analysis of the fluctuations of the condensate.

中文翻译：

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波动数量的独立随机变量的凝聚和极值

我们解决了三类密切相关的随机过程的凝聚和极端问题：（a）随机分配模型和零范围过程，（b）束缚更新过程，（c）自由更新过程。对于前一类，系统的组件数量是固定的，而对于其他两类，它是一个波动的数量。这些主题的研究分散在文献中，通常穿着其他服装。我们用独立随机变量的总和语言对该主题进行了精简说明，以便摆脱对特定模型的考虑并突出要点。除了对该理论进行统一介绍外，这项工作还调查了迄今为止在以前的研究中尚未探索的方面。具体来说，我们展示了对随机分配模型和零范围过程的研究如何作为研究当前工作核心的另外两类过程——束缚和自由更新过程的背景。然后，我们对这三类过程的极值统计提出了新的见解，从而可以更深入地了解凝结机理和凝结物波动的定量分析。