标题： 拓扑空间上的正则变化基础 显示英文标题

标题： Foundations of regular variation on topological spaces

摘要： 自J. Karamata引入以来，正则变化从一个纯粹的数学概念演变为理论概率和数据分析的基石。它在不同领域中得到了广泛的研究和应用。其重要性在于表征大偏差、确定部分和的极限以及预测随机过程中极端值的长期行为。受各种应用的启发，正则变化的框架随着时间的推移不断扩大，以包括更一般的空间中的随机观测，包括巴拿赫空间和波兰空间。在本专著中，我们确定了正则变化的三个基本组成部分：缩放、有界性和底层空间的拓扑结构。我们详细探讨了每个组成部分的作用，并将一些先前获得的结果扩展到一般的拓扑空间。我们更加抽象的方法统一了文献中出现的各种概念，简化了现有的证明，并为新的贡献铺平了道路，例如：随机测度和集合的（隐含）正则变化的广义理论；对由独立随机量缩放的正则变化随机函数和元素的创新处理以及许多其他进展。在全文中，关键结果和定义通过有指导意义的例子进行说明，包括对文献中几个已建立模型的扩展。通过将抽象与实用性相结合，这项工作旨在加深对正则变化的理论理解和方法适用性。 显示英文摘要

摘要： Since its introduction by J. Karamata, regular variation has evolved from a purely mathematical concept into a cornerstone of theoretical probability and data analysis. It is extensively studied and applied in different areas. Its significance lies in characterising large deviations, determining the limits of partial sums, and predicting the long-term behaviour of extreme values in stochastic processes. Motivated by various applications, the framework of regular variation has expanded over time to incorporate random observations in more general spaces, including Banach spaces and Polish spaces. In this monograph, we identify three fundamental components of regular variation: scaling, boundedness, and the topology of the underlying space. We explore the role of each component in detail and extend a number of previously obtained results to general topological spaces. Our more abstract approach unifies various concepts appearing in the literature, streamlines existing proofs and paves the way for novel contributions, such as: a generalised theory of (hidden) regular variation for random measures and sets; an innovative treatment of regularly varying random functions and elements scaled by independent random quantities and numerous other advancements. Throughout the text, key results and definitions are illustrated by instructive examples, including extensions of several established models from the literature. By bridging abstraction with practicality, this work aims to deepen both theoretical understanding and methodological applicability of regular variation.