PseudoRegularly Varying Functions and Generalized Renewal Processes
Authors: Buldygin, V.V., Indlekofer, K.H., Klesov, O.I., Steinebach, J.G.
Free Preview Introduces an interesting and quickly expanding theory
 Provides a rigorous treatment of the subject
 Presents a tool for application in financial mathematics and many other fields of applied mathematics and function analysis
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 About this book

One of the main aims of this book is to exhibit some fruitful links between renewal theory and regular variation of functions. Applications of renewal processes play a key role in actuarial and financial mathematics as well as in engineering, operations research and other fields of applied mathematics. On the other hand, regular variation of functions is a property that features prominently in many fields of mathematics.
The structure of the book reflects the historical development of the authors’ research work and approach – first some applications are discussed, after which a basic theory is created, and finally further applications are provided. The authors present a generalized and unified approach to the asymptotic behavior of renewal processes, involving cases of dependent interarrival times. This method works for other important functionals as well, such as first and last exit times or sojourn times (also under dependencies), and it can be used to solve several other problems. For example, various applications in function analysis concerning Abelian and Tauberian theorems can be studied as well as those in studies of the asymptotic behavior of solutions of stochastic differential equations. The classes of functions that are investigated and used in a probabilistic context extend the wellknown Karamata theory of regularly varying functions and thus are also of interest in the theory of functions.
The book provides a rigorous treatment of the subject and may serve as an introduction to the field. It is aimed at researchers and students working in probability, the theory of stochastic processes, operations research, mathematical statistics, the theory of functions, analytic number theory and complex analysis, as well as economists with a mathematical background. Readers should have completed introductory courses in analysis and probability theory.
 About the authors

Valerii Buldygin (1946–2012) received his PhD in Mathematics from Kiev Shevchenko University in 1973, and went on to work at the Institute of Mathematics, which is part of the Academy of Science of Ukraine (1973–1986) and at the National Technical University of Ukraine “Igor Sikorski Kyiv Polytechnic Institute” (1986–2012). His fields of research were probability and statistics with a special emphasis on limit theorems in Banach spaces, matrix normalizations in limit theorems, and asymptotic properties of correlograms and estimators of impulse transfer functions for linear systems.
KarlHeinz Indlekofer received his PhD in Mathematics from the Albert Ludwigs University of Freiburg im Breisgau in 1970. From 1970 to 1974 he worked at the Johann Wolfgang Goethe University of Frankfurt am Main. Since 1974 he has served as a Professor of Mathematics at the University of Paderborn. His field of research is analytic and probabilistic number theory with a special emphasis on sieve methods, limit theorems in number theory and arithmetical semigroups.
Oleg I. Klesov received his PhD in Mathematics from Kiev Shevchenko University in 1981. He worked at the same University as a scientific researcher until 1990, when he moved to the National Technical University of Ukraine “Igor Sikorski Kyiv Polytechnic Institute”. His fields of research include probability and stochastic processes with a special emphasis on random fields, limit theorems in probability, and the reconstruction of stochastic processes from discrete observations.Josef G. Steinebach received his PhD in Mathematics from the University of Dusseldorf in 1976. After a visit to Carleton University, Ottawa, in 1980, he worked as a Professor of Mathematics at the Universities of Marburg (1980–1987, 1991–2002), Hannover (1987–1991), and Cologne (since 2002). His field of research is probability and statistics with a special emphasis on changepoint analysis, limit theorems in probability, and asymptotic statistics.
 Table of contents (11 chapters)


Equivalence of Limit Theorems for Sums of Random Variables and Renewal Processes
Pages 125

Almost Sure Convergence of Renewal Processes
Pages 2752

Generalizations of Regularly Varying Functions
Pages 5397

Properties of Absolutely Continuous Functions
Pages 99151

Nondegenerate Groups of Regular Points
Pages 153199

Table of contents (11 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 PseudoRegularly Varying Functions and Generalized Renewal Processes
 Authors

 Valeriĭ V. Buldygin
 KarlHeinz Indlekofer
 Oleg I. Klesov
 Josef G. Steinebach
 Series Title
 Probability Theory and Stochastic Modelling
 Series Volume
 91
 Copyright
 2018
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer Nature Switzerland AG
 eBook ISBN
 9783319995373
 DOI
 10.1007/9783319995373
 Hardcover ISBN
 9783319995366
 Series ISSN
 21993130
 Edition Number
 1
 Number of Pages
 XXII, 482
 Number of Illustrations
 4 b/w illustrations
 Topics