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Advanced probability theory books?

  1. Oct 21, 2009 #1
    I'm interested in learning the calculus of general random variables, i.e. those that do not necessarily have a density or mass function - such as mixtures of continuous / discrete / Cantor-type variables.

    There seem to be several different approaches:

    1. Via densities, using delta functions etc, e.g. [tex]E[X]=\int_{-\infty}^{\infty}x f(x)dx[/tex]

    2. Via cumulative distributions, using Stieltjes-type integrals, e.g. [tex]E[X]=\int_{-\infty}^{\infty}xdF(x)[/tex]

    3. Via probability measures, e.g. [tex]E[X]=\int x d\mu(x)[/tex]

    Each seems to have a well developed rigorous theory. What would be the best approach to focus on, and what's a good accessible book on the subject?
     
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  3. Oct 21, 2009 #2

    jbunniii

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    If you learn (3), then you will develop (1) and (2) as part of the process. [itex]F(x) = \mu((-\infty,x])[/itex], and the probability density function [itex]f(x)[/itex] exists if [itex]F[/itex] is an absolutely continuous function.

    I like Billingsley's https://www.amazon.com/Probability-Measure-3rd-Patrick-Billingsley/dp/0471007102" because it's a very readable yet rigorous treatment that doesn't assume that you already know measure theory and Lebesgue(-Stieltjes) integration.
     
    Last edited by a moderator: May 4, 2017
  4. Oct 22, 2009 #3
    Thanks - sounds like measure theory is the way to go and will be useful for more advanced topics.

    Having a very basic and incomplete knowledge of Lebesgue integration, I'm tossing up between Billingsley and Shiryaev's https://www.amazon.com/Probability-Graduate-Texts-Mathematics-v/dp/0387945490/" - the gist of the reviews seems to be that B is more of a gentle essay-style introduction whereas S is more concise and efficiently organized. Any thoughts on this?
     
    Last edited by a moderator: Apr 24, 2017
  5. Oct 22, 2009 #4

    jbunniii

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    I haven't read Shiryaev's book, so I can't compare the two. Billingsley isn't organized as a reference; he deliberately interleaves the probability material with measure theory on an "as-needed" basis, which is nice because everything seems properly motivated as you read through it. I would not say that his book is gentle per se (parts of it are quite tough), but it flows pretty well and he does a good job letting you know what he's doing and why.

    Besides Billingsley and Shiryaev, another commonly used probability book at this level is Chung's https://www.amazon.com/Course-Proba...sr_1_1?ie=UTF8&s=books&qid=1256271371&sr=8-1". I've only skimmed it, and it looks fine, but a lot more dry than Billingsley. For example, Billingsley has a cool chapter about gambling theory, and often sprinkles interesting side topics such as "Strange Euclidean Sets" and the Banach-Tarski paradox, but Chung takes more of a no-nonsense approach. Chung is probably more appropriate for a graduate course, whereas Billingsley seems better for self-study. Just my opinion.
     
    Last edited by a moderator: Apr 24, 2017
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