Foundations of regular variation on topological spaces
Bojan Basrak, Nikolina Milinčević, Ilya Molchanov
TL;DR
The paper builds a unified, highly abstract framework for regular variation in general topological spaces by decomposing tail behaviour into scaling (group action), boundedness (ideals/bornologies), and topology. It develops vague convergence and α-homogeneous tail measures, extends polar (modulus) and transversal decompositions, and provides general continuous-mapping and quotient-space results. The framework encompasses metric, Banach, and non-mmetric contexts, enabling generalised tail theories for random measures, functions, sequences, and sets, including hidden regular variation and Breiman-type results. Practical implications include streamlined proofs and adaptable tail analyses across diverse spaces, from random variables to random closed sets and point processes. The work thus broadens the reach of regular variation to non-Euclidean settings, facilitating rigorous tail characterisation in complex stochastic models.
Abstract
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.