This second edition of a classic textbook on probability theory is designed for first-year graduate students who already have a solid foundation in undergraduate probability. It offers a comprehensive exploration of key concepts, enhancing the reader's understanding of advanced topics in probability. The updated content ensures relevance and clarity, making it an essential resource for students looking to deepen their knowledge in this fundamental area of mathematics.
From the reviews: „This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. [...] This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view.“ Mathematische Operationsforschung und Statistik
This book provides a rigorous but elementary introduction to the theory of Markov Processes on a countable state space. It should be accessible to students with a solid undergraduate background in mathematics, including students from engineering, economics, physics, and biology. Topics covered are: Doeblin's theory, general ergodic properties, and continuous time processes. Applications are dispersed throughout the book. In addition, a whole chapter is devoted to reversible processes and the use of their associated Dirichlet forms to estimate the rate of convergence to equilibrium. These results are then applied to the analysis of the Metropolis (a.k.a simulated annealing) algorithm. The corrected and enlarged 2nd edition contains a new chapter in which the author develops computational methods for Markov chains on a finite state space. Most intriguing is the section with a new technique for computing stationary measures, which is applied to derivations of Wilson's algorithm and Kirchoff's formula for spanning trees in a connected graph.
The book equips probabilists with the foundational knowledge necessary to effectively apply partial differential equations (PDEs) within probability theory and vice versa. It bridges the gap between these two fields, enabling readers to explore the interconnections and practical applications of PDEs in probabilistic contexts.
The book focuses on the fundamental theory of the Fourier transform, utilizing Norbert Wiener's original approach. It explores key concepts such as Parseval's formula and the Fourier inversion formula through the lens of Hermite functions. This edition enhances the learning experience with new exercises and solutions, making it a valuable resource for understanding advanced mathematical concepts related to Fourier analysis.
Rooted in a course from the Massachusetts Institute of Technology, this book delves into advanced concepts and methodologies relevant to its subject matter. It offers a unique blend of theoretical insights and practical applications, making it an essential resource for students and professionals alike. The content is designed to enhance understanding and foster critical thinking, providing readers with the tools needed to navigate complex topics effectively.
Focusing on Gaussian measures, this text serves as an introduction for students in a one-semester course. It highlights the role of Fourier analysis in probabilistic results and explores Gaussian measures in both finite and infinite dimensional spaces. Key properties that are dimension independent are emphasized, and Gaussian processes are constructed. The book also delves into Gaussian measures on Banach spaces, adopting insights from I. Segal and L. Gross. It is designed for those with a background in Lebesgue integration and functional analysis, offering techniques applicable beyond Gaussian measures.
Focusing on mathematical analysis, the book bridges advanced calculus and higher-level analysis, addressing both real and complex variables. It emphasizes Riemann integration in higher dimensions, providing a thorough exploration of Fubini's theorem, polar coordinates, and the divergence theorem. The final chapter utilizes these concepts to derive Cauchy's formula, demonstrating fundamental properties of analytic functions. This comprehensive approach equips readers with essential tools and ideas in mathematical analysis.
‘A Concise Introduction to the Theory of Integration’ was once a best-selling Birkhäuser title which published 3 editions. This manuscript is a substantial revision of the material. Chapter one now includes a section about the rate of convergence of Riemann sums. The second chapter now covers both Lebesgue and Bernoulli measures, whose relation to one another is discussed. The third chapter now includes a proof of Lebesgue's differential theorem for all monotone functions. This is a beautiful topic which is not often covered. The treatment of surface measure and the divergence theorem in the fifth chapter has been improved. Loose ends from the discussion of the Euler-MacLauren in Chapter I are tied together in Chapter seven. Chapter eight has been expanded to include a proof of Carathéory's method for constructing measures; his result is applied to the construction of Hausdorff measures. The new material is complemented by the addition of several new problems based on that material.
This book gives a somewhat unconventional introduction to stochastic analysis. Although most of the material coveredhere has appeared in other places, this book attempts to explain the core ideas on which that material is based. As a consequence, the presentation is more an extended mathematical essay than a ``definition, lemma, theorem'' text. In addition, it includes several topics that are not usually treated elsewhere. For example, Wiener's theory of homogeneous chaos is discussed, Stratovich integration is given a novel development and applied to derive Wong and Zakai's approximation theorem, and examples are given of the application ofMalliavin's calculus to partial differential equations. Each chapter concludes with several exercises, some of which are quite challenging. The book is intended for use by advanced graduate students and researchmathematicians who may be familiar with many of the topics but want to broaden their understanding of them.