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Generalizations of Regularly Varying Functions

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Book cover Pseudo-Regularly Varying Functions and Generalized Renewal Processes

Part of the book series: Probability Theory and Stochastic Modelling ((PTSM,volume 91))

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Abstract

Karamata [205] introduced the (so-called) regularly varying functions (RV-functions) in 1930 and proved a number of fundamental results for them (see also Karamata [206]). These results and their further generalizations formed the basis of a developed theory with a wide range of applications (cf., e.g., Seneta [324], Bingham et al. [41], Alsmeyer [11], Bingham [38], Marić [261], Korevaar [237], Resnick [300, 302], Yakymiv [373]). This chapter is devoted to the discussion of certain classes of functions which generalize, in one way or another, the notion of regular variation. Additionally to the classes RV and ORV of functions, an important place in the theory is occupied by the classes PRV, PI, SQI, and POV. The main attention in this chapter is paid to these four classes.

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Authors and Affiliations

  1. Department of Mathematical Analysis, National Technical University of Ukraine, Kyiv, Ukraine

    Valeriĭ V. Buldygin

  2. Department of Mathematics, University of Paderborn, Paderborn, Germany

    Karl-Heinz Indlekofer

  3. Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine, Kyiv, Ukraine

    Oleg I. Klesov

  4. Mathematical Institute, University of Cologne, Cologne, Germany

    Josef G. Steinebach

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  1. Valeriĭ V. Buldygin
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  3. Oleg I. Klesov
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  4. Josef G. Steinebach
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Buldygin, V.V., Indlekofer, KH., Klesov, O.I., Steinebach, J.G. (2018). Generalizations of Regularly Varying Functions. In: Pseudo-Regularly Varying Functions and Generalized Renewal Processes. Probability Theory and Stochastic Modelling, vol 91. Springer, Cham. https://doi.org/10.1007/978-3-319-99537-3_3

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