Geodesic
Karamata's Characterization Theorem, Feller, and Regular Variation in Probability Theory
Publications de l'Institut Mathématique, _N_S_71 (2002) no. 85, p. 79 .

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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.
DOI : 10.2298/PIM0271079S
Classification : 26A12 60F05 01A60 60-03
Keywords: characterization, boundedness on finite intervals, regularly varying functions, limit theorems in probability theory, necessary and sufficient conditions, central limit problem, history and biography 
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E. Seneta. Karamata's Characterization Theorem, Feller, and Regular Variation in Probability Theory. Publications de l'Institut Mathématique, _N_S_71 (2002) no. 85, p. 79 . doi : 10.2298/PIM0271079S. http://geodesic.mathdoc.fr/articles/10.2298/PIM0271079S/

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