PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 71(85), pp. 7989 (2002) 

KARAMATA'S CHARACTERIZATION THEOREM, FELLER, AND REGULAR VARIATION IN PROBABILITY THEORYE. SenetaSchool of Mathematics and Statistics FO7, University of Sydney, NSW 2006, AustraliaAbstract: 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;6003 Full text of the article:
Electronic fulltext finalized on: 19 Feb 2003. This page was last modified: 20 Feb 2003.
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