Essays on regular variations in classical and free setup: randomly weighted sums,products in CEVM and free subexponentiality

Abstract

In this thesis, we shall be focusing on some problems in probability theory involving regularly varying functions. The theory of regular variations has played an important role in probability theory, harmonic analysis, number theory, complex analysis and many more areas of mathematics. For an encyclopedic treatment of the subject, we refer to Bingham et al.(1987). In probability theory, the limiting behavior of the sums of independent and identically distributed (iid) random variables is closely related to regular variation. The books by Feller (1971) and Gnedenko and Kolmogorov (1968) give characterizations of random variables in the domains of attraction of stable distributions in terms of regularly varying functions. The study of extreme value theory was first initiated by Fisher and Tippett (1928), Gnedenko (1943). The use of regular variation in extreme value theory is now very well known due to the works of de Haan (1970, 1971). The Tauberian theorems involving regularly varying functions play a very crucial role in different disciplines. Bingham et al.(1987) gives a very good account of the Tauberian theorems for Laplace and Mellin transforms. The use of regularly varying functions is also very popular in insurance and risk theory. For a comprehensive study of different applications, we refer to the book by Embrechts et al.(1997).