It is expected that the candidate knows material from a standard undergraduate, post-calculus level course in probability:
For the exam, the student can choose one of two tracks.
Measure theoretic foundations of probability theory: probability spaces; random variables as measurable functions; notions of convergence (almost sure versus in probability)
Finite Markov chains in discrete time (recurrence vs. transience, periodicity, convergence to stationary distribution)
Markov chains with infinite state space: positive recurrence, null recurrence, and transience; reversible Markov chains; relationship between eigenvalues and rates of convergence to equilibrium; branching processes and random walks as examples
Poisson processes: definitions, thinning, superposition, conditioning
Markov chains with continuous time: infinitesimal generator; Kolmogorov equations for transition probabilities; relationship to embedded discrete time Markov chains. Birth and death chains and Markovian queues as examples
Brownian motion - definition and basic properties
Integration: Fatou's lemma, monotone and dominated convergence; product measures, Fubini's theorem
Probabilistic measure theory: Borel-Cantelli Lemmas; pi-lambda theorem, conditions for independence of events, random variables and sigma-fields; Kolmogorov extension theorem; Zero-One Laws (Kolmogorov and Hewitt-Savage)
Weak and strong laws of large numbers (proofs for finite variance); law of iterated logarithm (without proof)
Weak convergence of probability measures; characteristic functions of random variables and their relationship to weak convergence. Central limit theorem: be able to explain the ideas that underlie the proof for iid sequences
Conditional expectation. Martingales (in discrete time); upcrossing inequality, martingale convergence theoresm; Doob's inequality, Lp maximal inequality; uniform integrability; optional stopping theorem; applications to branching processes, Polya urns, Radon-Nikodym derivatives, etc.
Birkhoff ergodic theorem (without proof), Kac's recurrence theorem
Definition of Brownian motion; Kolmogorov continuity theorem; non-differentiability of paths; strong Markov property; reflection principle; Donsker's theorem
The student is allowed to exclude topics they are not comfortable with. However as in Olympic diving, the score on the qualifying exam will reflect both the difficulty of the material attempted and the quality of the performance.
A good way to review and broaden your knowledge to read another book on the subject.
Durrett. Essentials of Stochastic Processes.
Lawler. Introduction to Stochastic Processes.
Grimmett and Stirzaker. Stochastic Processes.
Liggett. Continuous Time Stochastic Processes. AMS.
Durrett. Probability Theory and Examples.
Rosenberg. A First Look at Rigorous Probability Theory.
Khoshnevisan. Probability. AMS
Athreya and Lahti. Measure Theoretic Probability Theory. Springer
Fristedt and Gray. A Modern Approach to Probability Theory. Birkhauser
Kallenberg. Foundations of Modern Probability Theory.