Rigorous probabilistic arguments, built on the foundation of measure theory introduced eighty years ago by Kolmogorov, have invaded many fields. 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 2002 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.
• Numerous exercises • Contains many comments, explanations and aids to intuition, not just wall-to-wall mathematics • Unusual treatment of advanced topics, using streamlined notation and methods accessible to students who have not studied probability at this level before
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.
'A really useful book …'. EMS Newsletter