I have a lot of respect for Nassim Taleb’s expertise in probability theory. I’ve gathered here some books on probability theory that have been recommended by Taleb, along with his commentary (when available).
- Feller, An introduction to probability theory and its applications, Vol. 1 & 2
Papoulis, Probability, Random Variables and Stochastic Processes
When readers and students ask to me for a useable book for nonmathematicians to get into probability (or a probabilistic approach to statistics), before embarking into deeper problems, I suggest this book by the Late A. Papoulis. I even recommend it to mathematicians as their training often tends to make them spend too much time on limit theorems and very little on the actual “plumbing”.
The treatment has no measure theory, cuts to the chase, and can be used as a desk reference. If you want measure theory, go spend some time reading Billingsley. A deep understanding of measure theory is not necessary for scientific and engineering applications; it is not necessary for those who do not want to work on theorems and technical proofs.
I’ve notice a few complaints in the comments section by people who felt frustrated by the treatment: do not pay attention to them. Ignore them. It the subject itself that is difficult, not this book. The book, in fact, is admirable and comprehensive given the current state of the art.
I am using this book as a benchmark while writing my own, but more advanced, textbook (on errors in use of statistical models). Anything derived and presented in Papoulis, I can skip. And when students ask me what they need as pre-requisite to attend my class or read my book, my answer is: Papoulis if you are a scientist, Varadhan if you are more abstract.
- Loeve, Probability Theory I & II
- Billingsley, Probability and Measure (Borel)
Varadhan, Probability Theory (Courant Lecture Notes)
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, Large Deviations (Courant Lecture Notes) (though older than this current text). The book on Stochastic Processes, Stochastic Processes (Courant Lecture Notes) should be paired with this one.
- Borel, Les probabilités dénombrables et leurs applications arithmétiques, 1909. For general intuition.
- Kolmogorov, On logical foundations of probability theory.
- Karatzas and Shreve, Brownian Motion and Stochastic Calculus
- Doob, Stochastic Processes
- Oksendal, Stochastic differential equations, 2013.
- Varadhan, Stochastic processes, 2007.
- Cover and Thomas, Elements of Information Theory
Extreme Value Theory
Embrechts et al., Modelling Extremal Events: for Insurance and Finance
The mathematics of extreme events, or the remote parts of the probability distributions, is a discipline on its own, more important than any other with respect to risk and decisions since some domains are dominated by the extremes: for the class of subexponential (and of course for the subclass of power laws) the tails ARE the story.
Now this book is the bible for the field. It has been diligently updated. It is complete, in the sense that there is nothing of relevance that is not mentioned, treated, or referred to in the text. My business is hidden risk which starts where this book stops, and I need the most complete text for that.
In spite of the momentous importance of the field, there is a very small number of mathematicians who deal with tail events; of these there is a smaller group who go both inside and outside the “Cramer conditions” (intuitively, thin-tailed or exponential decline).
It is also a book that grows on you. I would have given it a 5 stars when I started using it; today I give it 6 stars, and certainly 7 next year.
I am buying a second copy for the office. If I had to go on a desert island with 2 probability books, I would take Feller’s two volumes (written >40 years ago) and this one.
One housecleaning detail: buy the hardcover, not the paperback as the ink quality is weaker for the latter.
- De Haan and Fereira, Extreme Value Theory: An Introduction
- Gnedenko & Kolmogorov, Limit Distributions for Sums of Independent Random Variables
- Uchaikin & Zolotarev, Chance and Stability, Stable Distributions and Their Applications
- Samorodnitsky and Taqqu, Stable non-Gaussian random processes: stochastic models with infinite variance, 1994.
- Zolotarev, One-dimensional stable distributions, 1986.
- Pitman, Subexponential distribution functions, 1980.
- Embrechts and Goldie, On convolution tails, 1982.
- Embrechts et al., Subexponentiality and infinite divisibility, 1979.
- Chistyakov, A theorem on sums of independent positive random variables and its application to branching random processes, 1964.
- Goldie, Subexponential distributions and dominated-variation tails, 1978.
- Teugels, The class of subexponential distributions, 1975.
Franklin, The Science of Conjecture: Evidence and Probability before Pascal
Indispensable. As a practitioner of probability, I’ve read many book on the subject. More are linear combinations of other books and ideas rehashed without real understanding that the idea of probability harks back the Greek pisteuo (credibility) and pervaded classical thought. Almost all of these writers made the mistake to think that the ancients were not into probability. And most books such Bernstein, *Against the Gods* are not even wrong about the notion of probability: odds on coin flips are a mere footnote. If the ancients were not into computable probabilities, it was not because of theology, but because they were not into games. They dealt with complex decisions, not merely probability. And they were very sophisticated at it.
This book stands above, way above the rest: I’ve never seen a deeper exposition of the subject, as this text covers, in addition to the mathematical bases, the true philosophical origin of the notion of probability. In addition Franklin covers matters related to ethics and contract law, such as the works of the medieval thinker Pierre de Jean Olivi, that very few people discuss today.
Taleb has also remarked on Twitter that Stoyanov, Counterexamples in Probability is a good read.
A very good compilation of all of Taleb’s Amazon recommendations (as of 2012) is available on Farnam Street. Here I have limited the selections to probability theory alone; for mathematical finance, statistics, philosophy, etc., consult Farnum Street or Taleb’s Amazon page for the most recent reviews (e.g. Hastie, Elements of Statistical Learning and Goodfellow, Deep Learning).