KARAMATA'S CHARACTERIZATION THEOREM, FELLER, AND REGULAR VARIATION IN PROBABILITY THEORY

E. Seneta

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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