Regularly Varying Functions

The notion of regularly varying functions was introduced by Karamata in the 1930s, first in the Romanian journal Mathematica (Cluj) in 1930, and then in Bulletin de la Socit Mathmatique de France in 1933. Regularly varying functions appear to be a very useful tool in probability theory, especially in the theory of the summation of random variables, probabilistic extreme value theory, modeling of random heavy tail phenomena, and other important topics. In 1943, Gnedenko published his seminal paper in which he determined the domains of attraction of the Frchet and Weibull extreme value distributions in terms of regularly varying functions at infinity and at a fixed point, respectively. The importance of regularly varying functions in the theory of the summation of random variables was mention by Gnedenko and Kolmogorov 1949. Feller emphasized the importance of regularly varying functions in probability theory in his influential book An Introduction to Probability Theory and Its Applications. In 1970 de Haan introduced the classes of \(\varPi \) -varying and \(\varGamma \) -varying functions that play a crucial role in the extreme value theory together with regularly varying functions.