Topological regular variation: II. The fundamental theorems

This paper investigates fundamental theorems of regular variation (Uniform Convergence, Representation, and Characterization Theorems) some of which, in the classical setting of regular variation in R, rely in an essential way on the additive semi-group of natural numbers N (e.g. de Bruijn’s Representation Theorem for regularly varying functions). Other such results include Goldie’s direct proof of the Uniform Convergence Theorem and Seneta’s version of Kendall’s theorem connecting sequential de…nitions of regular variation with their continuous counterparts (for which see [BOst15]). We show how to interpret these in the topological group setting established in [BOst13] as connecting N-‡ow and R-‡ow versions of regular variation, and in so doing generalize these theorems to Rd: We also prove a ‡ow version of the classical Characterization Theorem of regular variation. Classi…cation: 26A03