Originating from the authors' own graduate course at the University of North Carolina, this material has been thoroughly tried and tested over many years, making the book perfect for a two-term course or for self-study. It provides a concise introduction that covers all of the measure theory and probability most useful for statisticians, including Lebesgue integration, limit theorems in probability, martingales, and some theory of stochastic processes. Readers can test their understanding of the material through the 300 exercises provided. The book is especially useful for graduate students in statistics and related fields of application (biostatistics, econometrics, finance, meteorology, machine learning, and so on) who want to shore up their mathematical foundation. The authors establish common ground for students of varied interests which will serve as a firm 'take-off point' for them as they specialize in areas that exploit mathematical machinery.
Ross Leadbetter is Professor of Statistics and Operations Research at the University of North Carolina, Chapel Hill. His research involves stochastic process theory and applications, point processes, and particularly extreme value and risk theory for stationary sequences and processes. Stamatis Cambanis was a Professor at the University of North Carolina, Chapel Hill until his death in 1995. He taught a wide range of statistics and probability courses and contributed very significantly to the development of the measure and probability instruction and the lecture notes on which this volume is based. Vladas Pipiras has been with the University of North Carolina, Chapel Hill since 2002 and rose to full Professor in 2012. His main research interests focus on stochastic processes exhibiting long-range dependence, multifractality and other scaling phenomena, as well as on stable, extreme-value and other distributions possessing heavy tails. He has also worked on statistical inference questions for reduced-rank models with applications to econometrics, and sampling issues for finite point processes with applications to data traffic modeling in computer networks.
Preface; Acknowledgements; 1. Point sets and certain classes of sets; 2. Measures: general properties and extension; 3. Measurable functions and transformations; 4. The integral; 5. Absolute continuity and related topics; 6. Convergence of measurable functions, Lp-spaces; 7. Product spaces; 8. Integrating complex functions, Fourier theory and related topics; 9. Foundations of probability; 10. Independence; 11. Convergence and related topics; 12. Characteristic functions and central limit theorems; 13. Conditioning; 14. Martingales; 15. Basic structure of stochastic processes; References; Index.