Probability with martingales

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1. chisquaredNov 2022
   David Williams’ Probability with Martingales is a personal favourite. And I’m not the only one.
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2. iflpOct 2020
   It's probably way easier to decide on a subject you want to learn more about, and find the most approachable textbook / lecture notes on e.g. StackExchange. It doesn't make sense to choose courses (or even subjects to learn) based on whether video lectures are available.

   I don't maintain a collection of these, but I can probably recommend

   * Stephen Abbott, Understanding Analysis, for mathematical analysis and a little bit of real analysis

   * David Williams, Probability with Martingales, for advanced probability

   * Dym and McKean, Fourier Series and Integrals

   Alternatively if you have a more cursory interest, you can probably check out Metacademy [1] which provides roadmaps to learning a variety of math (and ML) concepts.

   [1]: https://metacademy.org/browse

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3. dagwJan 2017
   On the probability front I recommend:

   - Probability and Random Processes by Grimmett

   - Probability with Martingales by David Williams

   Grimmett is probably a better bet to start with since it doesn't expect quite as much prior math knowledge and covers a lot more topics. The Williams book is shorter, denser and doesn't cover much in the way of applications, but gives a really good theoretical underpinning of how probability theorists think about probability (ie in terms of Lebesgue measures and Sigma algebras).

   edit: There is also a companion book to Probability and Random Processes called One Thousand Exercises in Probability which contains an interesting selections of problems and solutions that will let you apply the theories taught in the main book.

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4. nppOct 2010
   Advanced Linear Algebra, Roman; Linear Algebra, Hoffman & Kunze; Matrix Analysis, Horn & Johnson; Principles of Mathematical Analysis, Rudin; Real Analysis, Royden.

   Miscellaneous comments:

   - Reading pure abstract algebra (e.g. Dummit & Foote) isn't a good use of time if you intend to go into statistics, since it only shows up in a few very special subareas. If you decide to go into one of these areas, you can learn this later.

   - More advanced books on linear algebra usually emphasize the abstract study of vector spaces and linear transformations. This is fine, but you also need to learn about matrix algebra (some of which is in that Horn & Johnson book) and basic matrix calculus, since in statistics, you'll frequently be manipulating matrix equations. The vector space stuff generally does not help with this, and this material isn't in standard linear algebra books. (Similarly, you should learn the basics of numerical linear algebra and optimization -- convex optimization in particular shows up a lot in statistics.)

   - People have different opinions on books like Rudin, but you need to learn to read material like this if you're going into an area like probability. It's also more or less a de facto standard, so it is worth reading partly for that reason as well. So read Rudin/Royden (or equivalent, there are a small handful of others), but supplement them with other books if you need (e.g. 'The Way of Analysis' is the complete opposite of Rudin in writing style). It helps to read a few different books on the same topic simultaneously, anyway.

   - Two books on measure-theoretic probability theory that are more readable than many of the usual suspects are "Probability with Martingales" by Williams and "A User's Guide to Measure-Theoretic Probability" by Pollard. There is also a nice book called "Probability through Problems" that develops the theory through a series of exercises.

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5. dmvaldmanOct 2010
   The roadmap would be to first learn analysis (advanced calculus), then measure theory, then get a measure theory probability book.

   I'll assume you are teaching this to yourself.

   For analysis, I wouldn't get Rudin's book (concepts are poorly motivated). There are plenty of good Dover books. But I haven't read them because, well, I learned from Rudin.

   For measure theory, I'd read Kolmogorov and Fomin's book. Rudin also has a measure theory book, which is much better than his analysis book, but it's hefty. Good problems though.

   For a book on Probability, we read A Probability Path at my university. I wasn't fond of it. Someone referred me to Probability with Martingales, and though I didn't read it, it looked very good.

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