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 semigroup of natural numbers N (e.g. de Bruijnʹs Representation Theorem for regularly varying functions). Other such results include Goldieʹs direct proof of the Uniform Convergence Theorem and Senetaʹs version of Kendallʹs theorem connecting sequential definitions of regular variation with their continuous counterparts (for which see Bingham and Ostaszewski (2010) [13]). We show how to interpret these in the topological group setting established in Bingham and Ostaszewski (2010) [12] as connecting N -flow and R -flow versions of regular variation, and in so doing generalize these theorems to R d . We also prove a flow version of the classical Characterization Theorem of regular variation.