CDAM: Computational, Discrete and Applicable Mathematics@LSE

CDAM Research Report, LSE-CDAM-2008-12

August 2008

Topological regular variation: II. The fundamental theorems

N. H. Bingham and A. J. Ostaszewski

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 definitions 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-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.

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		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk