PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.)
Vol. 71(85), pp. 79--89 (2002)
KARAMATA'S CHARACTERIZATION THEOREM, FELLER, AND REGULAR VARIATION IN PROBABILITY THEORY
E. SenetaSchool of Mathematics and Statistics FO7, University of Sydney, NSW 2006, Australia
Abstract: Karamata's Characterization Theorem provided the impetus for Feller's (1966) exposition of the theory of regularly varying functions within a probability theory context. We investigate the conditions under which this theorem holds, and indicate manifestations in the identification of the spectral functions of the stable laws. Regular variation of a distribution function occurred implicitly as a necessary and sufficient condition for convergence in the 1930's, in the probabilistic work of P. Lévy, Khinchin, and Feller; and more transparently in that of Gnedenko and of Doeblin. Explicit recognition of the relevance of the concept in probability was interrupted by World War 2. A final section of this paper traces the evolution of Feller's name and early mathematical career from his Balkan origins, with a view to illuminating his recognition of the relevance of regular variation and his connection with Karamata.
Keywords: characterization; boundedness on finite intervals; regularly varying functions; limit theorems in probability theory; necessary and sufficient conditions; central limit problem; history and biography
Classification (MSC2000): 26A12; 60F05;01A60;60-03
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Electronic fulltext finalized on: 19 Feb 2003. This page was last modified: 20 Feb 2003.
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