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Probability Theory (Courant Lecture Notes), by S. R. S. Varadhan
Download Probability Theory (Courant Lecture Notes), by S. R. S. Varadhan
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This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation. In the first part of the book, characteristic functions are introduced, followed by the study of weak convergence of probability distributions. Then both the weak and strong limit theorems for sums of independent random variables are proved, including the weak and strong laws of large numbers, central limit theorems, laws of the iterated logarithm, and the Kolmogorov three series theorem. The first part concludes with infinitely divisible distributions and limit theorems for sums of uniformly infinitesimal independent random variables. The second part of the book mainly deals with dependent random variables, particularly martingales and Markov chains. Topics include standard results regarding discrete parameter martingales and Doob's inequalities. The standard topics in Markov chains are treated, i.e., transience, and null and positive recurrence. A varied collection of examples is given to demonstrate the connection between martingales and Markov chains. Additional topics covered in the book include stationary Gaussian processes, ergodic theorems, dynamic programming, optimal stopping, and filtering. A large number of examples and exercises is included. The book is a suitable text for a first-year graduate course in probability.
- Sales Rank: #372218 in Books
- Published on: 2001-09
- Original language: English
- Number of items: 1
- Dimensions: 10.25" h x 7.25" w x .50" l, .70 pounds
- Binding: Paperback
- 167 pages
Most helpful customer reviews
14 of 14 people found the following review helpful.
A gem
By N N Taleb
I know which books I value when I end up buying a second copy after losing the first one. This book gives a complete overview of the basis of probability theory with some grounding in measure theory, and presents the main proofs. It is remarkable because of its concision and completeness: visibly prof Varadhan lectured from these notes and kept improving on them until we got this gem. There is not a single sentence too many, yet nothing is missing.
For those who don't know who he is, Varadhan stands as one of the greatest probabilists of all time. Learning probability from him is like learning from Aristotle.
Varadhan has two other similar volumes one covering stochastic processes the other into the theory of large deviations (though older than this current text). The book on Stochastic Processes should be paired with this one.
14 of 15 people found the following review helpful.
The distilled core - an ideal first text for measure theoretic probability
By vidyut
This is an ideal text for first year graduate probability. Take Billingsley, which is several times bulkier, and distill the core concepts that one should away from the course, and that is Varadhan. Beautifully written, a joy to read, and occasionally even a useful reference to have on the desk to go back to when formalizing proofs for publication.
For brevity's sake though, quite a few ideas have been left to exercises, and I am not sure if this is a text that an uninitiated can learn from without having the luxury of being in a class. These are course notes that Varadhan used to teach the course himself, so probably the intended audience is students taking a course - not self study.
4 of 5 people found the following review helpful.
Very succint
By Yair Daon
Varadhan is very succint, doesn't give too many details. I didn't use this book too much. My lecturer was Amir Dembo and his notes are pretty good (exact opposit of Varadhan - much longer and go over all details and more).
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