Pseudo-Regularly Varying Functions and Generalized Renewal Processes

1. Front Matter

   Pages i-xxii

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2. Equivalence of Limit Theorems for Sums of Random Variables and Renewal Processes

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 1-25

3. Almost Sure Convergence of Renewal Processes

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 27-52

4. Generalizations of Regularly Varying Functions

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 53-97

5. Properties of Absolutely Continuous Functions

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 99-151

6. Nondegenerate Groups of Regular Points

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 153-199

7. Karamata’s Theorem for Integrals

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 201-228

8. Asymptotically Quasi-inverse Functions

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 229-310

9. Generalized Renewal Processes

   • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

   Pages 311-343

10. Asymptotic Behavior of Solutions of Stochastic Differential Equations

    • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

    Pages 345-393

11. Asymptotics for Renewal Processes Constructed from Multi-indexed Random Walks

    • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

    Pages 395-417

12. Spitzer Series and Regularly Varying Functions

    • Valeriĭ V. Buldygin, Karl-Heinz Indlekofer, Oleg I. Klesov, Josef G. Steinebach

    Pages 419-438

13. Back Matter

    Pages 439-482

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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 inter-arrival 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 well-known 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. 

• 60K05, 60F15, 26A12, 60H10, 60F10, 60E07
• generalized renewal processes
• pseudo-regular variation
• first and last exit time
• Sojourn time
• asymptotic behavior
• pseudo-regularly varying functions
• ordinary differential equations

• Department of Mathematical Analysis, National Technical University of Ukraine, Kyiv, Ukraine

  Valeriĭ V. Buldygin

• Department of Mathematics, University of Paderborn, Paderborn, Germany

  Karl-Heinz Indlekofer

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

  Oleg I. Klesov

• Mathematical Institute, University of Cologne, Cologne, Germany

  Josef G. Steinebach

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. 
Karl-Heinz 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 change-point analysis, limit theorems in probability, and asymptotic statistics.

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