Synopses & Reviews
This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.
Review
"Unlike technical books of a previous generation, here we have an author admitting that a reader might find the subject difficult and even offering a window on the pedagogical considerations by which he shapes his exposition. Pollard does not just explain and clarify abstractions; he really sells them to a presumably skeptical reader. Thus he bridges a gap in the literature, between elementary probability texts and advanced works that presume a secure prior knowledge of measure theory...The nice layout and occasional useful diagram further amplify the friendliness of this book." Choice
Review
"The book ... can be recommended as an excellent source in measuring theoretic probability theory as well as a handbook for everybody who studies stochastic processes in the real world." Mathematical Reviews
Synopsis
A secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.
Synopsis
This book offers a rigorous probability course for a mixed audience - statisticians, biostatisticians, mathematicians, economists, and students of finance - at the advanced undergraduate/introductory graduate level, without measure theory as a prerequisite. It covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms plus more advanced topics.
Synopsis
Rigorous probabilistic arguments, built on the foundation of measure theory introduced seventy years ago by Kolmogorov, have invaded many fields. Many students of statistics, biostatistics, econometrics, finance, and other changing disciplines now find themselves needing to absorb theory beyond what they might have learned in the typical undergraduate, calculus-based probability course. This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students, who were expected only to have taken an undergraduate course in real analysis or advanced calculus.
Table of Contents
1. Motivation; 2. A modicum of measure theory; 3. Densities and derivatives; 4. Product spaces and independence; 5. Conditioning; 6. Martingale et al; 7. Convergence in distribution; 8. Fourier transforms; 9. Brownian motion; 10. Representations and couplings; 11. Exponential tails and the law of the iterated logarithm; 12. Multivariate normal distributions; Appendix A. Measures and integrals; Appendix B. Hilbert spaces; Appendix C. Convexity; Appendix D. Binomial and normal distributions; Appendix E. Martingales in continuous time; Appendix F. Generalized sequences.