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I'm looking for a problem book in the style of Zhang's Linear Algebra: Challenging Problems for Students to prepare for a probability qualifying exam. In particular, the desirable source must have:

  • Graudate level difficulty (for a "just OK" school).
  • Complete, detailed solutions to problems.
  • Comprehensive coverage of probability fundamentals.

To clarify "comprehensive coverage," here is a subset of the topics:

  • Measure-theoretic foundations (Lebesgue construction, Dynkin/$\pi$-$\lambda$, DC/MCT/FL, Fubini/Tonelli, Radon-Nikodym, etc.)
  • Conditional probabilities/expectation
  • Convergences (weak, strong, probability, distribution, Borel-Cantelli, etc.)
  • Laws of large numbers (weak, strong)
  • Central limit theorems
  • Discrete time stochastic processes (martingales, branching processes, Doob decomposition, Markov, recurrence, etc.)
  • Continuous time stochastic processes
  • Brownian Motion (Wiener measure [which always makes me giggle], tightness, strong Markov, etc.)

For greater detail, the following textbooks were used for the associated classes:

I care more about being adept at solving probability problems (which I currently am not) than about passing this particular exam, so good references that do not necessarily cover some of this material are still desirable, where "good" means the inclusion of clear, well-written solutions to provide feedback.

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"One Thousand Exercises in Probability" by Grimmett and Stirzaker is a possible suggestion, though INMHO not as promising as it sounds. Some pros and some cons:

  • pros: covers many areas and $1000$ exercises are a $1000$ exercises! Graduate level: check $\checkmark$
  • cons: does not have the, let's say standard, exercises in each subject, so INMHO it serves better as a complement to a textbook that covers - through examples or exercises - the basics (or more) in every subject.

(My) conclusion: It will certainly help you but keep looking around.


For books with clear, well-written solutions, you could also check Hoel Port Stone "Introduction to Probability Theory" and Bertsekas, Tsitsiklis "Introduction to Probability 2nd Edition". The solutions may also be found online (for sure for the second one). But these cover more basic subjects.

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