#### Date of Submission

4-28-2011

#### Date of Award

4-28-2012

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Maulik, Krishanu (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

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 (i.i.d.) 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). The studyof record values and record times also uses various properties of extreme value theory and regularly varying functions (Resnick, 1987, Chapter 4). The theory of multivariate regular variation is useful in modelling various telecommunication systems and Internet traffic (Heath et al., 1999, Maulik et al., 2002, Mikosch et al., 2002). The study of certain environmental issues is also facilitated when one considers the theory of multivariate regular variations. See de Haan and de Ronde (1998), Heffernan and Tawn (2004) for some recent applications in this area.The theory of regular variations has also found an important place in free probability. Free probability was introduced by Voiculescu (1986). While free probability has an important application in the study of random matrices, its connection with various topics like the study of the free groups and the factor theory have made it a subject of its own interest. Bercovici and Pata (1999, 2000b) studied the domain of attraction of free stable laws and showed that the regularly varying functions play a very crucial role like they do in the classical setup. In the final chapter of this thesis, we show another application of regular variation in free probability theory.In Section 1.2 we give the definitions of regularly varying functions, distribution functions and measures with regularly varying tails and some of their properties. In Subsections 1.2.1 and 1.2.2 we state some well known results on the regularly varying functions which we shall use in the later chapters. In Section 1.3 we recall the definitions of subexponential and long tailed distributions and some of their properties. In Section 1.4 we give a brief introduction to one dimensional extreme value theory and also point out its connections with regularly varying functions. In Section 1.5 we give a brief overview of the results in the later chapters.

#### Control Number

ISILib-TH305

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Hazra, Rajat Subhra Dr., "Essays on Regular Variations in Classical and Free Setup: Randomaly Weighted Sums, Products in CEVM and Free Subexponentiality." (2012). *Doctoral Theses*. 40.

https://digitalcommons.isical.ac.in/doctoral-theses/40

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842816