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I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems fabulous, but it has way too much content on Measure Theory, which I do not know at all.

There is a wealth of professors and undergrad students of Stats and Maths on this website; and I would like to directly ask you: which book should suit my situation? Because I CAN learn Measure Theory but considering I am only a first year undergrad, and Measure Theory will be taught to me three years later, I suppose it will only be a wastage of time.

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    $\begingroup$ Personally, I feel there's few things as useful as learning some measure theory as early as possible, especially if you're seriously interested in probability; but that's just my opinion. $\endgroup$
    – Noah Schweber
    Commented Jan 23, 2015 at 17:48
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    $\begingroup$ This site is focused only on research level questions, so textbook recommendations for undergrads are off topic. math.stackexchange.com would be a better place. But the rigorous definitions of the various modes of convergence of random variables are all in terms of measure theory, so I don't think you can really study this subject properly without understanding measure theory. I agree with @NoahS that it wouldn't be a waste of your time at all (provided you have the necessary background now); you'll just find it much easier in three years. $\endgroup$
    – Nate Eldredge
    Commented Jan 23, 2015 at 18:06
  • $\begingroup$ Also, entirely separately - and I should have mentioned this in my first comment - measure theory is incredibly cool. See it now so you can see more later! $\endgroup$
    – Noah Schweber
    Commented Jan 23, 2015 at 18:13
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    $\begingroup$ While I agree that concepts of convergence can be understood only via measure theory, one can get to a certain degree a good understanding of limit theorems without it. E.g., Emanuel Lesigne's "Heads or Tails" (ams.org/bookstore-getitem/item=STML-28) is a very readable introduction. $\endgroup$
    – Stephan Sturm
    Commented Jan 23, 2015 at 19:16

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Grimmett and Stirzaker's Probability and Random Processes fits the bill:

http://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220

It covers everything up to stochastic calculus without measure theory.

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    $\begingroup$ I like the book very much. But the convergence of random variables appears only in Chapter 7 and uses measure theory. Only the weakest form of convergence (in distribution) does not require measure theory. $\endgroup$
    – Liviu Nicolaescu
    Commented Jan 23, 2015 at 19:16
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    $\begingroup$ @LiviuNicolaescu: I was actually thinking that what's presented in chapter 7 is not really measure theory (no sigma-algebras, simple-functions, measures, etc). Just the modes of convergence defined. The truth is I read the book after taking a class on measure theory so the concepts were of course familiar. But again, I would think that this is as good as OP will find if they want to use a book that presents "graduate" level probability with basically no mention of measures. $\endgroup$
    – Alex R.
    Commented Jan 23, 2015 at 19:51

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