Probability Theory

10 Oct 2024 10:09

One of my advisers in graduate school was a probability theorist, as was his adviser before him; I've not bothered to check, but I wouldn't be astonished if the chain went back to someone like Bernoulli. The fact that the chain could go back that far shows that mathematical probability is an old concept, almost as old as any other part of modern science; on the other hand, my adviser's adviser came just after the generation, between the wars, which made probability a respectable and rigorous branch of mathematics and removed countless obscurities from its applications. The first serious use of statistical methods in the sciences came only about a hundred years before that. Now of course error analysis is the first thing my students learn when they enter the lab. (Well, almost the first thing, after "if you don't write it down, it never happened" and "Cosma can be bribed with chocolate.") I am conditioned to attack every problem as some kind of stochastic process; but a few generations back nobody had any but the vaguest idea what a stochastic process was.

Pet peeves: Physicists who do not distinguish between a random variable ("X = the roll of a die") and the value it takes ("x=5"). People who report estimated numbers without error-bars or confidence-intervals. Bayesians.

Patrick Billingsley, Probability and Measure
Harald Cramér, Mathematical Methods of Statistics [Review]
William Feller, An Introduction to Probability Theory and Its Applications, vol. I [I've not finished vol. II yet...]
Bert Fristedt and Lawrence Gray, A Modern Approach to Probability Theory [Extremely thorough measure-theoretic text; nice treatment of stochastic processes]
Geoffrey Grimmett and David Stirzaker, Probability and Random Processes [Maybe the best contemporary textbook for those who do not need measure-theoretic probability]
Ian Hacking
- The Emergence of Probability [Where that strange two-faced notion came from, and why]
- The Taming of Chance [Putting chance to work in the 19th century]
Mark Kac, Probability and Related Topics in Physical Science
Olav Kallenberg, Foundations of Modern Probability [My preferred textbook when teaching stochastic processes]
Michel Loève, Probability Theory
David Pollard, A User's Guide to Measure-Theoretic Probability
R. F. Streater, "Classical and Quantum Probability," math-ph/0002049 ["There are few mathematical topics that are as badly taught to physicists as probability theory."]
Aram Thomasian, The Structure of Probability Theory

Philippe Barbe, "An Elementary Approach to Extreme Values Theory", arxiv:0811.0753
Jochen Brocker, "A Lower Bound on Arbitrary f-Divergences in Terms of the Total Variation" arxiv:0903.1765
H. E. Daniels, "Mixtures of Geometric Distributions", Journal of the Royal Statistical Society B 23 (1961): 409--413 [JSTOR]
Alexander E. Holroyd and Terry Soo, "A Non-Measurable Set from Coin-Flips", math.PR/0610705 [A cute construction to help students see the point of measure-theoretic probability]
Mark Kac
- Selected Papers
- Statistical Independence in Probability, Analysis and Number Theory
Olav Kallenberg, Probabilitic Symmetries and Invariance Principles [A tremendous book, but I must admit to a disappointment. The three basic symmetries Kallenberg considers are symmetry under permutation (exchangeability), symmetry under rotation, and symmetry under "contraction" (i.e., integrating out variables). The obvious fourth is symmetry under translation, or stationarity; this he frankly skips, on the grounds that so much has been written about it elsewhere. But I would very much have liked to read his approach to it...]

Clark Glymour, "Instrumental Probability", Monist 84 (2001): 284--300 [PDF reprint]
Mark Kac, Engimas of Chance: An Autobiography
Jill North, "Symmetry and Probability", phil-sci/2978
Oystein Ore, Cardano, the Gambling Scholar
Aris Spanos, "A frequentist interpretation of probability for model-based inductive inference", Synthese 190 (2011) [With thanks to Prof. Spanos for letting me read a pre-publication draft]
Jakob Rosenthal, "The Natural-Range Conception of Probability", phil-sci/4978 [Defends the thesis that "the probability of an event is the proportion of initial states that lead to this event in the space of all possible initial states, provided that this proportion is approximately the same in any not too small interval of the initial state space.... [I]n the types of situations that give rise to probabilistic phenomena we may expect to find an initial state space such that any 'reasonable' density function over this space leads to the same probabilities for the possible outcomes."]

Lorraine Daston, Classical Probability in the Enlightenment
Gerd Gigerenzer, Zeno Switjtink, Theodore Porter, Lorraine Daston, John Beatty and Lorenz Krüger, The Empire of Chance: How Probability Changed Science and Everyday Life
Kendall and Plackett (eds.), Studies in the History of Statistics and Probability
Andrei Kolmogorov, Foundations of Probability Theory
Francesco Mainardi, Sergei Rogosin, "The origin of infinitely divisible distributions: from de Finetti's problem to Levy-Khintchine formula", arxiv:0801.1910
Glenn Shafer and Vladimir Vovk, "The Sources of Kolmogorov's Grundbegriffe", Statistical Science 21 (2006): 70--98 = math.ST/0606533
Reinhard Siegmund-Schultze
- "Probability in 1919/20: the von Mises-Pólya-Controversy", Archive for History of Exact Sciences 60 (2006): 431--515
- "A Non-Conformist Longing for Unity in the Fractures of Modernity: Towards a Scientific Biography of Richard von Mises (1883--1953)", Science in Context 17 (2004): 333--370
Jan von Plato, Creating Modern Probability [I need to finish this, one of these years...]

Marshall Abrams, "Toward a Mechanistic Interpretation of Probability", phil-sci/4704
Jordan Ellenberg and Elliott Sober, "Objective Probabilities in Number Theory" [PDF preprint]
Eduardo M. R. A. Engel, A Road to Randomness in Physical Systems
John L. Pollock, Nomic Probability and the Foundations of Induction
Alexander R. Pruss, "Probability, Regularity, and Cardinality", Philosophy of Science 80 (2013): 231--240
John T. Roberts, "Laws About Frequencies", phil-sci/5058
Michael Strevens, Bigger than Chaos: Understanding Complexity through Probability
Jon Williamson, Lectures on Inductive Logic

Blom, Holst and Sandell, Problems and Snapshots from the World of Probability ["It is obvious that the authors have had fun in writing this book..."]
Feller, An Introduction to Probability Theory and Its Applications vol. II [I need to finish one of these decades...]
Svante Janson, "Probability asymptotics: notes on notation", arxiv:1108.3924 [Looks useful for the next time I teach stochastic processes]
Olav Kallenberg, Foundations of Modern Probability, 3rd edition [As I say above, the 2nd edition has long been my standard text when teaching measure-theoretic probability and stochastic processes, and I am genuinely eager to read the new material here]
Emmanuel Lesigne, Heads or Tails: An Introduction to Limit Theorems in Probability
Sidney Resnick, A Probability Path
A. Shiryaev, Probability Theory
Jordan Stoyanov, Counterexamples in Probability Theory
D. Stroock, Probability Theory: An Analytic View
Paul Vitanyi, "Randomness," math.PR/0110086

David J. Aldous and Antar Bandyopadhyay, "A survey of max-type recursive distributional equations", math.PR/0401388 = Annals of Applied Probability 15 (2005): 1047--1110
David Balding, Pablo A. Ferrari, Ricardo Fraiman and Mariela Sued, "Limit theorems for sequences of random trees", math.PR/0406280 [Abstract: "We consider a random tree and introduce a metric in the space of trees to define the "mean tree" as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we show laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law." I wonder if the same technique could be applied to other kinds of random graphs, e.g., random scale-free networks?]
Patrick Billinglsey, Convergence of Probability Measures
Salomon Bochner, Harmonic Analysis and the Theory of Probability
Tapas Kumar Chandra, The Borel-Cantelli Lemma
Louis H. Y. Chen, Larry Goldstein and Qi-Man Shao, Normal Approximation by Stein's Method
Sourav Chatterjee, "A new method of normal approximation", arxiv:math/0611213
Bernard Chazelle, The Discrepency Method: Randomness and Complexity
Irene Crimaldi and Luca Pratelli, "Two inequalities for conditional expectations and convergence results for filters", Statistics and Probability Letters 74 (2005): 151--162
Victor De La Pena and Evarist Gine, Decoupling: From Dependence to Independence
Victor H. de la Pena, Tze Leung Lai and Qi-Man Shan, Self-Normalized Processes: Limit Theory and Statistical Applications
Janos Galambos and Italo Simonelli, Bonferroni-type Inequalities with Applications
Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas, "A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties", arxiv:0711.1478
J. A. Gonzalez, L. I. Reyes, J. J. Suarez, L. E. Guerrero, and G. Gutierrez, "A mechanism for randomness," nlin.CD/0202022 [Color me skeptical, from the abstract]
Martin Hairer, "A theory of regularity structures", arxiv:1303.5113
Laurent Mazliak, "Poincarés Odds", arxiv:1211.5737
Henry McKean, Probability: The Classical Limit Theorems
National Research Council (USA), Probability and Algorithms [online]
Peter Orbanz, "Projective limit random probabilities on Polish spaces", Electronic Journal of Statistics 5 (2011): 1354--1373
Iosif Pinelis, "Between Chebyshev and Cantelli", arxiv:1011.6065
Chris Preston, "Some notes on standard Borel and related spaces", arxiv:0809.3066
Revesz, The Laws of Large Numbers
R. Schweizer and A. Sklar, Probabilistic Metric Spaces
Glenn Shafer and Vladimir Vovk, Probability and Finance: It's Only a Game! [Yet Another Foundation of Probability, this time from game-theory.]
Akimichi Takemura, Vladimir Vovk, Glenn Shafer, "The generality of the zero-one laws", arxiv:0803.3679
Ramon van Handel, Probability in High Dimension [PDF lecture notes]
Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science