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These lecture notes are intended for a first-year graduate-level course on measure-theoretic probability. Topics covered include: foundations, independence, zero-one laws, laws of large numbers, weak convergence and the central limit theorem, conditional expectation, martingales, Markov chains and Brownian motion. The notes were used for two semester courses at UW-Madison (Spring 2018, Fall 2013) and two quarter courses at UCLA (Winter 2012, Winter 2011, Fall 2010, Winter 2010). The material is based largely on the following references:

- [D] Probability: Theory and Examples by Durrett
- [W] Probability with Martingales by Williams
- [S] Probability by Shiryaev
- [B] Probability and Measure by Billingsley
- [MP] Browninan Motion by Morters and Peres
- [R] Modern Discrete Probability: An Essential Toolkit by Roch

The notes are divided into (roughly) one week worth of material (two 75-minute lectures):

- Notes 1: Measure-theoretic foundations I
- Notes 2: Measure-theoretic foundations II
- Notes 3: Modes of convergence
- Notes 4: Laws of large numbers
- Notes 5: More on the a.s. convergence of sums
- Notes 6: First and second moment methods
- Notes 7: Concentration inequalities
- Notes 8: Weak convergence and characteristic functions
- Notes 9: CLT and Poisson Convergence
- Notes 10: Method of moments
- Notes 11: Infinitely divisible and stable laws
- Notes 12: Random walks
- Notes 13: Conditioning
- Notes 14: Martingales
- Notes 15: Branching processes
- Notes 16: Martingales in Lp
- Notes 17: UI Martingales
- Notes 18: Optional sampling theorem
- Notes 19: Martingale CLT
- Notes 20: Azuma's inequality
- Notes 21: Markov chains: definition, properties
- Notes 22: Markov chains: stationary measures
- Notes 23: Markov chains: asymptotic behavior
- Notes 24: Markov chains: martingale methods
- Notes 25: Ergodic theory: brief introduction
- Notes 26: Brownian motion: definition
- Notes 27: Brownian motion: path properties
- Notes 28: Brownian motion: Markov property
- Notes 29: Brownian motion: martingale property

Last updated: August 16, 2022.