Sidney Resnick: Books
2007 Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer-Verlag, New York.
This comprehensive text gives an interesting and useful blend of the mathematical, probabilistic and statistical tools used in heavy-tail analysis. Heavy tails are characteristic of many phenomena where the probability of a single huge value impacts heavily. Record-breaking insurance losses, financial-log returns, files sizes stored on a server, transmission rates of files are all examples of heavy-tailed phenomena. Numerous examples and exercises drive the exposition and the mathematical properties of the methodologies as well as their implementation in Splus or the R statistical languages is discussed. Both probability modeling and statistical methods for fitting models receive consideration.
1987 Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New York. New Springer paperback printing, 2008.
This is a monograph describing the mathematical underpinnings of Extreme Value Theory. There are two lines of development, both of which are useful for deep understanding of extremes. The analytical basis for the subject is the theory of regularly varying functions. The probabilistic basis for deep understanding is the theory of point processes. These are linked through the idea that regular variation of a probability tail is equivalent to convergence of induced point processes to a limiting Poisson process. This is a surprisingly powerful and economical idea which provides a probabilistic transform.
Contents: Preface.- Preliminaries.- Domains of Attraction and Norming Constants.- Quality of Convergence.- Point Processes.- Records and Extremal Processes.- Multivariate Extremes.- References.- Index
Note: This is reprinted and reissued by Springer as a paperback in 2008.
1992 Adventures in Stochastic Processes. Birkhauser, Boston.
This textbook provides easy access to stochastic processes for students of applied science at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is very accessible to beginners but flexible enough to serve those who come to the course with strong backgrounds. Adventures in Stochastic Processes contains many examples, exercises, and applications of a practical and serious nature. Underlying principles of complex problems and computations are cleanly and quickly delineated through lively and rich vignettes of a personalized group of characters inhabiting our random world. Each chapter is fully supplemented with exercises.
Condensed contents: Preliminaries: Discrete Index Sets and/or Discrete State Spaces * Markov Chains * Renewal Theory * Point Processes * Continuous Time Markov Chains * Brownian Motion * The General Random Walk * Index
1998 A Probability Path. Birkhauser, Boston. Also available as a paperback in the series Modern Birhauser Classics.
Many probability books are written by mathematicians and have the built in bias that the reader is assumed to be a mathematician coming to the material for its beauty. This textbook is geared towards beginning graduate students from a variety of disciplines whose primary focus is not necessarily mathematics for its own sake. Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance, and engineering. A one-semester course is laid out in an efficient and readable manner covering the core material. The first three chapters provide a functioning knowledge of measure theory. Chapter 4 discusses independence, with expectation and integration covered in Chapter 5, followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. Two subsequent chapters offer a careful treatment of convergence in distribution and the central limit theorem. The final chapter treats conditional expectation and martingales, closing with a discussion of two fundamental theorems of mathematical finance. Like Adventures in Stochastic Processes, A Probability Path is rich in appropriate examples, illustrations, and problems, and is suitable for classroom use or self-study.
2001 Levy Processes, Theory and Applications (volume edited with O. Barndorff-Nielsen, T. Mikosch). Birkhauser, Boston.
A LÚvy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. The need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general LÚvy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of LÚvy processes and their enormous flexibility in modeling tails, dependence and path behavior.
This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. It collects articles written by leading experts that will appeal to the non-specialist. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text.
For the researcher and graduate student, every article contains open problems and points out directions for future research. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of LÚvy processes.