Slowly varying functions in the Zygmund sense and generalized regular variation

Abstract

The fundamental role of slowly varying functions in the Zygmund sense is examined. We derive a necessary and sufficient condition for a positive measurable function to be slowly varying in the Zygmund sense. Analytical equivalences involving Zygmund SVF's in probabilistic setting are related to necessary and sufficient analytical conditions for classical limit theorems. In this context an analytical conjecture is presented. Finally, the role of Zygmund SVF's in recent generalizations of regular variation is presented as a unifying principle.