Foundations of Regular Variation on Topological Spaces

Since its introduction by J. Karamata, regular variation has evolved from a purely mathematical concept into a cornerstone of theoretical probability and statistical data analysis. It has been extensively studied and applied in different areas. Its significance lies in characterising tail behaviour, 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 elements in more general spaces, notably 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 previous results to general topological spaces. Our abstract approach unifies various concepts appearing in the literature, streamlines existing proofs, and paves the way for novel contributions. These include a generalised theory of (hidden) regular variation for random measures and sets and a treatment of regularly varying random functions and elements scaled by independent random quantities, among other developments. Throughout the text, key results and definitions are illustrated by instructive examples, including extensions of several established models. By bridging abstract theory and practice, this work aims to deepen both the theoretical understanding and the methodological applicability of regular variation.