## The Syllabus for the Qualifying Exam in Probability and Stochastic Processes

It is expected that the candidate knows material from a standard undergraduate, post-calculus level course in probability:

• Basic properties of probability and conditional probability including Bayes rule
• Discrete probability densities (binomial, Poisson, geometric, hypergeometric)
• Continuous probability densities (normal, exponential, uniform)
• Joint, marginal, and conditional densities
• Expectation, variance, standard deviation, covariance
• Poisson approximation to binomial
• Chebyshev's inequality and weak law of large numbers
• Central limit theorem

For the exam, the student can choose one of two tracks.

• Track I, only for students who will not write their thesis in probability, consists primarily of stochastic processes from a non-measure theoretic perspective and corresponds to Mathematics 541.
• Track II consists primarily of measure theoretic probability as taught in Statistics 711 and/or Mathematics 641.

### Core Material (Required for either Track)

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)

### Track I

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

### Track II

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

### Notes

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.

### References

Core Material

Pitman. Probability

Track 1.

Durrett. Essentials of Stochastic Processes.

Lawler. Introduction to Stochastic Processes.

Grimmett and Stirzaker. Stochastic Processes.

Liggett. Continuous Time Stochastic Processes. AMS.

Track 2.

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.