Topological regular variation: II. The fundamental theorems |
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Authors: | N.H. Bingham |
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Affiliation: | a Mathematics Department, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom b Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom |
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Abstract: | 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 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 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 Rd. We also prove a flow version of the classical Characterization Theorem of regular variation. |
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Keywords: | Multivariate regular variation Uniform convergence theorem Topological dynamics Flows |
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