Welcome! Here is some basic information regarding Advanced Probability.

Lecture notes and example sheets can be downloaded here (in pdf format).

• Lecture notes (corrected version, 14 January 2013).
• Example sheet 1
  Solutions.
• Example sheet 2
  Solutions.
• Example sheet 3
  Solutions.
• Example sheet 4
  Solutions.

Material from previous years:

• Perla Sousi's notes from 2011.
  We will be following these closely for the first 2 chapters.
• James Norris' notes
  These contain some material, like Markov chains and stochastic optimality, that we will not touch upon.

Miscellaneous material:

• Physical Brownian motion, after Brown and Einstein.
• Splendors and miseries of martingales (feat. J.L. Doob, P.A. Meyer, P. Levy, et al.)
• Levy processes in nature; hungry sharks as random walkers with heavy tails.

Our progress in the lectures will be charted here:

1. Lecture October 4: Material covered:
   • Sigma-algebras, measures, random variables.
   • Borel sigma-algebras, and sigma-algebras generated by RV's.
   • Expected value. Conditioning wrt single events. Conditional expectation in the discrete case.
   Corrections:
   • Sloppy definition of $L^{\infty}$ to be amended.
2. Lecture October 6:
   • L^p spaces. L^2 as Hilbert space, orthogonal projection.
   • Conditional expectation, general case: as projection in L^2 case, extension to integrable RVs using truncation.
   • Properties of conditional expectation. Proof of "taking out what is known".
3. Lecture October 9:
   • Tower property of conditional expectation.
   • Product sigma-algebras and product measures. Fubini-Tonelli theorem.
   • Conditional density functions.
   Corrections:
   • In pi-system uniqueness, finiteness of the measures was not explicitly stated as requirement.
4. Lecture October 11:
   • Filtrations and adapted processes.
   • Definition and examples of martingales. Sub/supermartingales.
   • Definition and basic examples of stopping times.
5. Lecture October 13:
   • Adaptedness of stopped processes; stopped processes as martingales.
   • Optional stopping theorem for bounded stopping times.
   • Counterexample with unbounded stopping time for random walk.
6. Lecture October 16:
   • Random walks, sample paths, gambler's ruin.
   • Doob's maximal and $L^p$ inequalities, with proofs.
7. Lecture October 18:
   • Martingale convergence theorem: statement, discussion, and proof.
   • Doob's upcrossing lemma, with proof.
8. Lecture October 20:
   • $L^p$ converge for martingales. Representation in terms of closed martingales.
   • $L^1$ convergence and uniform integrability.
   • Applications: tail-$\sigma$-algebras, random harmonic series.
9. Lecture October 23:
   • Kolmogorov's 01 law. Convergence of random series.
   • Backwards martingales.
   • Strong law of large numbers. Proof using backwards martingales.
   Corrections:
   • In the proof of Kolmogorov's theorem on random series, we should argue on
     $\P(\sup_{N\leq k\leq M}|S_k-S_N|>\epsilon)$ rather than
     $\P(\sup_{N\leq k\leq M}|S_k|>\epsilon)$.
     See lecture notes.
10. Lecture October 25:
    • Extension of definitions to continuous time: Martingales, filtrations, etc.
    • Sample path properties. Continuous and cadlag processes.
    • Progressive measurability. Stopping times for continuous and cadlag processes.
11. Lecture October 27:
    • Stopped cadlag processes.
    • Versions of processes, indistinguishability of processes, finite-dimensional distributions.
    • Remarks concerning canonical processes.
12. Lecture October 30:
    • Martingale regularization, assuming the usual conditions.
    • Poisson process. Construction. Prop's of increments. Compensated PP as martingale in continuous time.
13. Lecture November 1:
    • Kolmogorov's continuity criterion.
    • Weak convergence. First definition, and first examples.
14. Lecture November 3:
    • Weak convergence, general definition.
    • Characterizations of weak convergence.
    • Characteristic functions (Fourier transforms).
15. Lecture November 6:
    • More on convergence in distribution.
    • Sequences of distribution functions. Tightness.
    • Prokhorov's theorem. Proof in the case $M=\mathbb{R}$.
16. Lecture November 8:
    • Levy's continuity theorem, with proof.
    • Central limit theorem, with main steps of proof.
17. Lecture November 10:
    • Preliminary observations concerning large deviations.
    • Moment generating functions and their properties.
18. Lecture November 13:
    • Cram'er's theorem on large deviations, with proof.
    • Definition of Brownian motion. Invariance prop's. Markov property.
    • Almost sure non-differentiability of BM at a point.
19. Lecture November 15:
    • Construction of BM via fdd's, Daniell-Kolmogorov thm, and Kolmogorov's criterion.
    • Hölder continuity of BM.
    • Blumenthal's 01-law. BM is positive/hits zero immediately.
20. Lecture November 17:
    • Martingales and BM.
    • BM as strong Markov process.
    • Reflection principle. Running maximum. Passage time densities.
21. Lecture November 20:
    • Martingales from BM using Markov prop and heat eq.
    • Recurrence/transcience of d-dimensional BM.
    • Proof of nghbd-recurrence in 2-dim case.
22. Lecture November 22:
    • Kakutani's probabilistic solution to the Dirichlet problem.
    • Skorokhod embedding theorem.
23. Lecture November 24:
    • Donsker's invariance principle, with proof.
    • BM plus indep compensated Poisson processes.
    • Heuristic discussion of the Levy-Khinchin formula.
24. Lecture November 27:
    • Levy processes. Levy-Khinchin formula.
    • Poisson random measures, integrals with respect to PRM's.
    • Compensated PRM's and cadlag martingales.

Thanks for participating in the course!

Contact: a.sola(no_spam_please)statslab.cam.ac.u