Synopses & Reviews
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.
Review
"This book is different from the classical textbooks on probability theory in that it treats the measure theoretic background not as a prerequisite but as an integral part of probability theory. The result is that the reader gets a thorough and well-structured framework needed to understand the deeper concepts of current day advanced probability as it is used in statistics, engineering, biology and finance... The pace of the book is quick and disciplined. Yet there are ample examples sprinkled over the entire book and each chapter finishes with a wealthy section of inspiring problems." --Publications of the Int'l Statistical Institute "A very nice introductory measure-theoretic probability textbook... The emphasis [is] on developing a broad, deeper understanding of basic probability theory, the needs of students kept specifically in mind... Students will appreciate its clear, concise, and focused presentation of the material." --JASA "This book is a rather comfortable probability path leading from fundamental notions of measure theory and probability theory to martingale theory, since fundamental concepts and basic theorems are always illustrated by important examples and results of additional theoretical value." --Statistics & Decision "This textbook offers material for a one-semester course in probability, addressed to students whose primary focus is not necessarily mathematics...Each chapter is completed by an exercises section. Carefully selected examples enlighten the reader in many situations. The book is an excellent introduction to probability and its applications." ---Revue Roumaine de Mathématiques Pures et Appliquées
Synopsis
This text is geared towards students who have no prior exposure to measure theory or advanced probability, and who require this material for their work in statistics, applied probability, operations research, mathematical finance, and engineering.
A one semester path is laid out in an efficient and disciplined way to cover 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. This is followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. The remaining chapters offer a careful treatment of convergence in distribution and the central limit theorem. Finally, the text treats conditional expectation and martingales, closing on the fundamental theorem of mathematical finance.
Table of Contents
1. Sets and Events ; 2. Probability Spaces; 3. Random Variables, Elements and Measurable Maps; 4. Independence; 5. Integration and Expectation; 6. Convergence Concepts; 7. Laws of Large Numbers and Sums of Independent Random Variables; 8. Convergence in Distribution; 9. Characteristic Functions and the Central Limit Theorem; 10. Martingales; Index; References