I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in Mathematics - as possible. What I mean by that is that the book should contain and rigorously address the following topics:
I think Amir Dembo's notes are pretty stellar. He updates them each time he teaches the course, but even then they have really good proofs and exercises. He also has notes on stochastic processes. William's Probability with Martingales is also good... but only the parts on martingales are good. Durrett's book is decent.
A Course In Probability Weiss Pdf
I am of the persuasion that stochastic processes should be done in depth as its own course, and for the Oksendal "Stochastic Differential Equations" is easier and more insightful than Karatzas "Brownian Motion and Stochastic Calculus" which is tougher but more thorough, together making a good combo. I came out the other side still wanting to learn Malliavin calculus (still haven't gotten around to it) but feeling ready to do so.
There's a strange problem with probability and stats textbooks where the notation of explanation is exceptionally shoddy and non-rigorous. This book usually doesn't suffer from that deficiency. But occasionally it punts on topics that would require a familiarity with measure-theoretic concepts.
I'm a professor at UCLA. My research interests lie in descriptive set theory and its connections to related areas such as computability theory, combinatorics, ergodic theory, probability, operator algebras, and quantum information.
Fall 2020: MATH 285D, MIP*=RE The course will be an introduction to quantum information theory, with thegoal of covering the recent result of Ji, Natarajan, Vidick, Wright, andYuen that MIP*=RE: the class of languages which can be decided by a multiprover interactive protocol with a classical polynomial-time verifier and provers sharing arbitrarily many entangled qbits is equal to the class of all recursively enumerable languages. This implies a negative solution to the longstanding Connes embedding problem in operator algebras.
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