Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Suggested readings for students in probability

Inspired by greater people than me (Landau, 't Hooft, etc.,) I made a list of basic material that aspiring probabilists may find useful. It is not complete in any sense, because I lack knowledge of many sub-areas of probability and when there are multiple books covering the same material, I mention just one. Thanks to Yogeshwaran for many good suggestions that I have incorporated in making the list. In addition to these, one could also make a list of good books in analysis, theoretical CS, statistical mechanics but someone else should make those lists...

Most basic

  • Feller Probability theory and its applications- vol. 1
  • John B. Walsh Knowing the odds
One should know the material in these books, end to end. Many alternate books are possible, for example Probability: Theory and examples. For learning measure theory and probability, one could use Dudley's Analysis and probability while Grimmett and Stirzaker's Probability and random processes is good for pre-measure theoretical probability. Also, solving problems from the above books or from this set, for example is recommended. Kallenberg's inimitable book Foundations of probability contains material worth 5-6 other books, but my feeling is that it is best for the second reading rather than the first.

Basic, but slightly more advanced

Some of the topics that are basic but not adequately covered in the Durrett's book can be read from the following list. This is a topic-wise listing, rather than a recommendation to read all the books cover to cover!

  • Weak convergence theory from Billingsley's Weak convergence of probability measures or K. R. Parthasarathy's Probability measures on metric spaces.

  • Stochastic Calculus (Karatzas and Shreve Brownian motion and stochastic calculus)
  • Large deviations, at least Sanov's theorem (Dembo and Zeitouni's Large deviations: techniques and applications) and concentration of measure (first three chapters of Boucheron, Lugosi and Massart's Concentration inequalities: a non-asymptotic theory of independence).
  • Discrete probability: The amazingly beautiful book Probability on trees and networks by Lyons and Peres, Grimmett's Probability on graphs and the superb lecture notes by Sebastain Roch and the well-known book The probabilistic method by Alon and Spencer.
  • More about Brownian motion, its connections to PDE etc. (Varadhan's TIFR lecture notes Diffusions and partial differential equations)
  • Markov processes (Liggett's Continuous time Markov processes: an introduction, just the first 3-4 chapters would be good)
  • Mixing times of Markov chains (book of the same name by Levin, Peres and Wilmer, a few chapters give sufficiently many of the ideas)
  • Gaussian Hilbert spaces (Janson's Gaussian Hilbert spaces, initial part, is very good for learning Wiener chaos decomposition, Wick formula etc. It is also good to study stationary Gaussian processes on the line, and some basics of continuity of Gaussian processes)
  • Basics of information theory (Cover and Thomas have written a good book. Basic notions of entropy, relative entropy etc are essential for probabilists)
  • Point processes, random measures. (Two upcoming books: Kallenberg's Random measures, theory and applications which may be an update of his old book titled Random measures and Lectures on the Poisson Process by Last and Penrose. The first 9 chapters of the latter appear to be an adequate introduction to the basics)

Advanced topics

I exclude books already mentioned above. For almost any topic in probability, one can find some volume of the St. Flour lecture notes by an expert. In some cases, these may have become a little outdated, but as introductions, they are reliable (although the readability is highly variable!).

  • Mörters and Peres Brownian motion (for everything you want to know on sample path properties of Brownian motion)
  • Michel Ledoux Concentration of measure phenomenon (the other book mentioned earlier is easier, but this is already a classic)
  • Garban and Steif Noise sensitivity and percolation (Also published as a book, this gives a good exposition of the increasingly important notions of influence, noise sensitivity etc.).
  • Talagrand Upper and lower bounds for stochastic processes (reading it feels like getting a privileged view of a great mind)
  • Bass Probabilistic Techniques in Analysis
  • Bogachev Gaussian processes, Adler and Taylor Random Fields and Geometry, Lifshits Lectures on Gaussian Processes (one will need to learn about Gaussian processes at some point).
  • Normal approximations with Malliavin Calculus: From Stein's method to universality by Nourdin and Peccati
  • Remco van der Hofstad Random graphs and complex networks (another hot area, but introduced from scratch)
  • Grimmett Percolation (the standard reference for this area of probability)
  • Liggett Interacting particle systems (the standard reference for this area - see also Liggett's other book mentioned above)
  • Rogers and Williams Diffusions, markov processes and martingales
  • Panchenko Sherrington-Kirkpatrick model (Extremely well-written book on a difficult and important current area of research)
  • Anderson, Guionnet and Zeitouni An introduction to random matrices, Pastur and Shcherbina Eigenvalue Distribution of Large Random Matrices, Forrester Log-Gases and Random Matrices, Tao Topics in random matrix theory (Random matrices are currently very fashionable, these books and the next two are almost pairwise disjoint in their contents!)
  • Vershynin A course in high dimensional probability, Hopcroft and Kannan As yet nameless book on high dimensional probability
  • Steele Probability theory and combinatorial optimization (charmingly written, deliciously short!). Similar books in more geometric settings are Penrose's Random geometric graphs and Yukich's Probability Theory of Classical Euclidean Optimization Problems.
  • Lawler and Limic Random walk: a modern introduction
  • Lawler Conformally invariant processes in the plane (the unique book for learning about SLE, the single most important development in probability in recent times)
  • Chatterjee Super-concentration and related topics (Specialized topic but clearest possible exposition of some recent advances)
  • Villani Topics in optimal transportation (superb exposition, clear as a crystal!)
  • Diaconis Group representations in probability and statistics (who else will you learn it from?)
  • Aldous Probability Approximations via the Poisson Clumping Heuristic (a mostunusual book reflecting the originality of the author. If nothing else read the preface and a couple of applications)
  • Dan Romik The surprising mathematics of longest increasing subsequences (presents the story of a simply stated combinatorial problem that led mathematicians through combinatorics, probability, analysis, representation theory, algebraic geometry,...)

For those with a more historical inclination

  • Mark Kac's books, Probability and related topics in the physical sciences, Statistical independence in probability, analysis and number theory, Enigmas of chance (the last is an autobiography, but very nice to read. May also recommend here Ulam's Adventures of a mathematician). If I knew enough French, I would try to read Lévy's autobiography Quelques aspects de la pensée d’un mathématicien.
  • The Berkeley symposia on mathematical statistics and probability are very interesting to browse.
  • Selected works of A. N. Kolmogorov, vol. 2 is on the grandmaster's works in probability and statistics.
  • I have found it useful to browse through some original papers of Wiener (on Brownian motion), Kac (on precursors to Donsker's theorem), etc. One could perhaps also browse collected works of other famous probabilists.
  • A very interesting article by Shafer and Vovk on the world of probability up to Kolmogorov's definitive introduction of the axiomatic foundations.
  • Fischer A history of the central limit theorem

Particularly for those working with me

Books not included above