Summer Term 2020

lecture Prof. Dr. Nicolas Perkowski, exercise joint with Helena Kremp

---

Time and place

•  Lecture: Video lectures will be available online (see below).
  Online "office hours": Tuesdays 12:30-14:00. The link to the Webex session can be found in the Whiteboard system.

• Exercise Session: Tuesdays, 16:00-17:30, online. The link to the Webex session can be found in the Whiteboard system.
• Final Exam:  to be announced in due course

---

Prerequisits: Stochastics I und Analysis I — III.

Assessment

To receive credits fo the course you need to

• actively participate in the exercise session 
• work on and successfully solve the weekly exercises 
• pass the final exam (see above)

If you are an FU student you only need to register for the course via CM (Campus Management).
If you are not an FU student, you are required to register via KVV/Whiteboard.   

Exercises 

Problem sets will be put online every Tuesday and can be found under Assignements in the KVV/Whiteboard portal. Solutions (in groups of three!) are due by 4pm on Tuesday of the following week  –  solutions must be uploaded through the KVV/Whiteboard portal or sent by email to Helena Kremp.

Course Overview/ Content:

• Construction of stochastic processes;
• conditional expectation;
• martingales in discrete time: convergence, stopping theorems, inequalities;
• convergence types in stochastics;
• uniform integrability;
• Markov chains in discrete and continuous time: recurrence and transience, invariant measures;
• convergence in distribution of stochatic processes;
• Brownian motion and invariance principle.

References

• Klenke: Wahrscheinlichkeitstheorie
• Durrett: Probability. Theory and Examples.

Further literature will be given during the lecture.

Lecture videos

• Videos for Tuesday, April 21:
  • Welcome
  • Introduction (Section 0 of the notes)
  • Definition of stochastic process (1.1-1.2)
  • Distribution of a stochastic process (1.3-1.5)
  • Finite-dimensional distributions of a stochastic process (1.6-1.7)
• Videos for Thursday, April 23:
  • Definition and Examples of Polish spaces (1.8-1.10)
  • Tightness and regularity of probability measures (1.11-1.12)
  • Probabilites on Polish spaces are tight (1.13-1.15)
• Videos for Tuesday, April 28:
  • Consistency; Statement of Kolmogorov's extension theorem (1.16-1.18)
  • Proof of Kolmogorov's extension theorem (1.18)
• Videos for Thursday, April 30:
  • Construction of product measures & Gaussian processes (1.19-1.20)
  • First example of a Markov chain (1.21)
  • Definition of a Markov chain (1.22-1.26)
  • Construction of Markov chains (1.27-1.28)
• Videos for Tuesday, May 5:
  • A motivating example for the conditional expectation (2.1)
  • Lebesgue spaces (2.2)
  • Orthogonal projections in Hilbert spaces (2.3)
  • Properties of orthogonal projections (2.4-2.5)
• Videos for Thursday, May 7:
  • Definition of the conditional expectation (2.6-2.7)
  • Construction of the conditional expectation (2.7)
  • Conditional expectation for discrete sigma algebras (2.8-2.9)
  • Conditional expectation with joint density (2.10-2.12)
  • Key properties of conditional expectations (2.13)
• (Short) Videos for Tuesday, May 12:
  • Proof of key properties of conditional expectations (2.13)
  • Convergence theorems for conditional expectations (2.14)
  • Regular conditional probabilities & Jensen's inequality (2.16-2.18)
  • Integrating out independent influences; Wald's inequalities (2.19-2.20)
• Videos for Thursday, May 14:
  • Filtrations and martingales (3.1-3.3)
  • Examples of martingales (3.5-3.6)
  • Martingale transforms (3.7-3.8)
  • Convex functions of martingales (3.11-3.12)
• Videos for Tuesday, May 19:
  • Stopping times (3.13-3.15)
  • Stopping theorem (3.16-3.20)
  • Events observable until a stopping time (3.21-3.22)
  • Optional sampling theorem (3.24-3.26)
• Videos for Tuesday, May 26:
  • Applications of the optional sampling theorem (3.27-3.28)
  • Doob's upcrossing inequality (3.29-3.32)
  • Martingale convergence theorem (3.29)
• Videos for Thursday, May 28:
  • First applications of the martingale convergence theorem (3.33-3.35)
  • A positive martingale converging to zero (3.36)
  • The main convergence types of stochastics, I (3.37-3.40)
  • The main convergence types of stochastics, II (3.41-3.42)
  • The main convergence types of stochastics, III (3.43-3.46)
• Videos for Tuesday, June 2:
  • Uniform integrability (3.47-3.49)
  • Criteria for uniform integrability (3.50-3.51)
  • Examples for uniform integrability (3.52)
  • L1 convergence and uniform integrability (3.53)
  • Extensions of Theorem 3.53 (3.54-3.55)
• Videos for Thursday, June 4:
  • Applications of uniform integrability to martingales, I (3.56)
  • Applications of uniform integrability to martingales, II (3.57-3.59)
  • Application of Kolmogorov's 0-1 law (3.60)
  • Optional sampling version theorem, II (3.61-3.62)
  • Doob's inequality (3.63)
  • Doob's Lp inequality (3.64)
• Videos for Tuesday, June 9:
  • Azuma's inequality (3.65-3.67)
  • Hoeffding's inequality and an application (3.68-3.69)
  • A "new" definition of Markov chains (4.1)
  • Basic aspects of Markov chains (4.2)
• Videos for Thursday, June 11:
  • Ehrenfest model, Galton-Watson process, PageRank process (4.3-4.5)
  • Chapman-Kolmogorov equation (4.6-4.7)
  • Stationary and reversible measures (4.8)
  • Statement of the fundamental theorem of Markov chains (4.9)
• Videos for Tuesday, June 16:
  • Examples of Markov chains with zero or many stationary measures (4.10-4.12)
  • Irreducibility and communication classes (4.13)
  • Recurrence, transience, strong Markov property (4.14-4.17)
  • Recurrence and transience of random walks (4.18-4.19)
  • Existence of stationary measures for recurrent chains (4.20)
• Videos for Thursday, June 18:
  • Existence / uniqueness of stationary measures (4.21-4.22)
  • Existence of stationary probability measures (4.23-4.25)
  • Aperiodicity (4.26-4.28)
  • Coupling (4.29-4.33)
  • Doubling of variables, Markov chain on SxS
  • Proof of the fundamental theorem of Markov chains
• Videos for Tuesday, June 23:
  • A brief discussion of the ergodic theorem (4.34-4.35)
  • The PageRank algorithm (5.1)
  • A brief discussion of the Perron-Frobenius theorem (5.2-5.3)
• Videos for Thursday, June 25:
  • Monte Carlo simulation (5.4)
  • Definition of the Ising model (5.5)
  • Random walk on a finite graph (5.6-5.10)
  • The Metropolis-Hastings Markov chain (5.11-5.15)
• Videos for Tuesday, June 30:
  • Ising model revisited (5.16)
  • Main definitions of Bayesian statistics (5.17-5.18)
  • Bayes's formula, Markov chain Monte Carlo (5.19-5.21)
  • Conjugate priors (5.22)
• Videos for Thursday, July 2:
  • Weak convergence (6.1-6.2)
  • Weak convergence vs convergence in probability (6.3-6.7)
  • The portmanteau theorem (6.8-6.10)
  • Weak convergence on Rd (6.11-6.12)
• Videos for Tuesday, July 7:
  • Tightness (6.13-6.17)
  • Rescaled random walks (6.18)
  • Multidimensional Gaussians (6.19-6.24)
  • Brownian motion (6.25-6.27)
• Videos for Thursday, July 9:
  • Towards weak convergence on the Wiener space (6.28-6.30)
  • The Wiener space (6.31-6.32)
  • Two sigma-algebras on the Wiener space (6.33)
  • Characterization of weak convergence on the Wiener space (6.34)
• Videos for Tuesday, July 14:
  • The Arzela Ascoli theorem (6.35-6.36)
  • Tightness on the Wiener space (6.37)
  • Statement of Donsker's invariance principle and a new tightness criterion (6.38-6.39)
• Videos for Thursday, July 16:
  • Tightness estimates for rescaled random walks (6.39-6.41)
  • Proof of Donsker's invariance principle
  • The Brownian motion (6.42, Section 6.6)