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On this website, there are many questions about books on probability theory, but I would like to ask if you can suggest a book (or more than one if necessary) that is:

  • rigorous and accurate according to modern standards;
  • complete: from basic concepts and ideas to really advanced material;
  • rich of useful applications of techniques and ideas of probability theory to other branches of mathematics and physics.
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    $\begingroup$ Tempting to answer with my favorite book , "An Introduction to Probability Theory and Its Applications" in two volumes, by William Feller. You probably want to look at something more modern like "Probability" by A. N. Shiryaev $\endgroup$ – Alan Oct 1 '14 at 0:15
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In my opinion, three books that meet the criteria listed in the OP are:

  • Probability Theory: The Logic of Science by E.T. Jaynes, 2002 version: this is a comprehensive book that illustrates probability theory from basic (first volume) to advanced concepts (second volume), combining a modern, rigorous approach with an enjoying style, often using interesting digressions that go beyond conventional mathematics, and analyzing a large number of applications of probability in physics, mechanics, and other fields. In my opinion, this probability textbook is surely one of the best choice for readers interested in applications and ideas, and who look for something more than conventional statistical books.

  • Applied probability by K. Lange, 2010: a very interesting textbook with a particolar focus on a variety of applications and examples taken from biology and computational techniques, written with a rigorous mathematical style. It starts from elementary probability theories and covers a large variety of applications of measure theory, cominatorics, expectations, optimization theory, and so on, extending to more advanced issues such as modern probabilistic approach to steady flow, Brownian motion, dynamics of porous media, population modeling, and many others. A detailed analysis of stochastic processes including Poisson and branching processes, Markov chains, martingales, and diffusion processes is also provided in a very clear manner and with several practical examples. In summary, this is another textbook suitable for those who search something "more" in the applications of probability theory.

  • Probability in Physics by Y. Ben-Menaheml and M. Hemmo, 2012: this is a nice high-level book included in the Springer "Frontier collection", a series of volumes dedicated to advanced, open issues at the forefront of modern science. The book is very well-structured, written in a readable and accessible manner, and covers a large range of issues and ideas of probability in physics, starting from very basic concepts such as notion of probability and chance in physics, tipicality, role of Lebesgue measures in statistic mechanics, and so on, and passing through a number of applications that include axioms of quantum mechanics, contemporary probabilistic approach to equilibrium, controversial relation between locality and determinism, modern view of Maxwell's demons, quantum decoherence, and many other. In synthesis, a third book that may be of interest for those who search mathematical rigour, applications and ideas.

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Feller's book was published in 1968. Mandelbrot said of Feller's illustration of the random walk ,"The curve looked like a mountain's profile or cross section, and the points where it intersects the time axis reminded me of certain records I was then investigating relative to telephone errors. " So other people have found the book inspiring. It is one of the most enjoyable books I have read.

Incidentally Chebyshev's inequality is on page 233 of Volume I. I'm using this fact to illustrate that there is less analysis then algebra in this first volume. By contrast , Chebyshev's Theorem is proved in Shirayaev's book on page 47, ( Apostol's book Calculus Volume II has a chapter on probability , ( Chapter 14) Chebyshev's inequality is proved on page 563. And I think a book on real analysis , ( Folland for instance) has a proof too, page 185) . So , be prepared for hard analysis if you pick up a copy of Shiryaev's book. Others can perhaps come up with more suggestions.

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A couple of interesting books that include applications in an original way, as compared to other probability and stochastic processes books, are the following:

  • Markov Chains Gibbs Fields, Monte Carlo Simulation and Queues, by Pierre Bremaud. Is a book mainly about applications on Markov Chains with a complete set of results in discrete time, including advanced perspectives not included in basics texts: coupling techniques, ergodicity via Lyapunov functions and martingales, and spectral theory. In applications it has specific sections and scattered reference through out the text. It contains the most standard applications such as queues, but it does provide an introduction to advanced topics like Poisson calculus. It contains applications to signal processing and simulation. The index can be found in the authors page:
    http://lcavwww.epfl.ch/~bremaud/SVcontents.ps

  • Applied Probability and Queues, by Soren Assmusen. This book is a jewel, it contains an advanced measure theoretic description of markov chains (both continuous and discrete time) an its applications to queueing systems. It contains advanced reference to factorizations of random walk (spitzer theorems), renewal theory is very well developed, and contains sections on large deviations, measure changes, Levy processes and reflected processes. These results are applied to simulation problems that stemmed from queuing theory or for not so conventional queueing problems.

These two are a great alternative for the classical literature, of which I recommend Feller's volumes.

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Two more excellent books to be considered are:

  1. Introduction to Probability by Bertsekas and Tsitsiklis is an excellent book for getting the fundamentals right and to think in terms of applications.

  2. Probability Theory by S.R.S.Varadhan is supposed to be very precise in its treatment.

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Try Chorin, Hald: Stochastic Tools in Mathematics and Science. It is based on a course at UC Berkeley which was designed following a survey of former students, where the basic question was: "What kind of stochastics you really need for your scientific work?"

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