### Syllabus

### A Sample Midterm

### Materials on Random Walks

### Lecture Slides

### Final (Due 05/10/22)

**Lecture 1****Lecture 2****Lecture of 02/02****Lecture of 02/04**### Homeworks

**Homework 1:**1.6.2, 1.7.3, 2.1.1, 2.1.9.**Due 02/07****Homework 2:**2.2.1, 2.2.2, 2.2.4, 2.2.5.**Due 02/14****Homework 3:**2.3.15, 2.3.18, 2.4.1, 2.4.2.**Due 02/21****Homework 4:**2.5.1, 2.5.2, 2.5.9, 2.5.10.**Due 02/28****Homework 5:**3.2.6, 3.2.12, 3.2.13, 3.2.14.**Due 03/23****Homework 6:**3.3.13, 3.4.5, 3.4.6, 3.4.12.**Due 03/30****Homework 7:**4.1.8, 4.1.9, 4.1.10, 4.1.11 from the file Materials on Random Walks.**Due 04/06****Homework 8:**4.1.5, 4.1.7, 4.1.8, and this problem**Due 04/18****Homework 9:**4.2.5, 4.2.9, 4.2.10, and this problem**Due 04/27****Homework 10:**4.3.11, 4.4.3, 4.4.6 and 4.4.7**Due 05/02**### Class Diary

**Wednesday, 01/19:**sigma-field, measurable space, measure, probability measure, measure space, probability space,measures on the real line, Stieltjes function**Read:**Section 1.1**Friday, 01/21:**measures on high dimensional Euclidean space, random variables, distributions, distribution functions**Read:**Sections 1.1 and 1.2**Monday, 01/24:**Random variables, integration**Read:**Sections 1.3 and 1.4**Wednesday, 01/26:**Integration**Read:**Section 1.4**Friday, 01/28:**Jensen's inequality and Holder inequality**Read:**Section 1.5**Monday, 01/31:**bounded convergence theorem, Fatou's lemma, monotone convergence theorem, expected values**Read:**Sections 1.5 and 1.6**Wednesday, 02/02:**Theorem 1.6.8, change of variable formula, \pi-system, \lambda-system, \pi-\lambda theorem, product measures, Fubini's theorem**Read:**Sections 1.6 and 1.7**Friday, 02/04:**Independence, sums of independent random variables**Read:**Section 2.1**Monday, 02/07:**Kolmogorov's extension theorem, L^2 weak law, polynomial approximation**Read:**Sections 2.1.4 and 2.2.1**Wednesday, 02/09:**Triangular arrays, coupon colelctor's problem, occupancy problem, weak law for triangular arrays, weak law of large numbers**Read:**Section 2.2., 2.2.3**Friday, 02/11:**weak law of large numbers, St.Petersburg paradox, Borel-Cantelli lemma**Read:**Sections 2.2.3, 2.3**Monday, 02/14:**L^2 strong law of large numbers, 2nd Borel-Cantelli lemma,**Read:**Section 2.3**Wednesday, 02/16:**head runs, strong law of large numbers, renewal theory,**Read:**Sections 2.3 and 2.4**Friday, 02/18:**empirical distribution, The Glivenko-Cantelli theorem, tail events, Kolmogorov's 0-1 law, Kolmogorov's maximal; inequality**Read:**Sections 2.4 and 2.5**Monday, 02/21:**Kolmogorov's 3 series theorem, rates of convergence**Read:**Section 2.5**Wednesday, 02/23:**Infinite mean**Read:**Section 2.5.2**Friday, 02/25:**Weak convergence, characterization of weak convergence,**Read:**Sections 3.1, 3.2.1 and 3.2.2**Monday, 02/28:**continuous mapping theorem, Portmanteau theorem, Helly's selection theorem, tightness**Read:**Section 3.2.2**Wednesday, 03/02:**tightness, characteristic functions, properties of characteristic functions, the inversion formula**Read:**Sections 3.2.2, 3.3.1**Friday, 03/04:**the inversion formula, continuity theorem**Read:**Sections 3.3.1 and 3.3.2**Monday, 03/07:**moments and derivatives, Polya's criterion**Read:**Sections 3.3.3 and 3.3.4**Wednesday, 03/09:**Centrla limit theorem**Read:**Section 3.4.1**Friday, 03/11:**Midterm**Monday, 03/21:**Lindeberg-Feller theorem, the converse of the three series theorem**Read:**Section 3.4.2**Wednesday, 03/23:**Infinite variance, rates of convergence**Read:**Section 3.4.2, 3.4.4, 2.5**Friday, 03/25:**permutable events, exchangeable sigma-field, Hewitt-Savage 0-1 law**Read:**Section 2.5 (or Section 4.1 in the file Materials on Random Walks)**Monday, 03/28:**stopping times and their applications**Read:**Section 4.1 in the file Materials on Random Walks.**Wednesday, 03/30:**Wald's equation, simple random walks, Wald's 2nd equation**Read:**Section 4.1 in the file Materials on Random Walks.**Friday, 04/01:**Applications of Wald's 2nd equation to random walks, recurrence, transience, simple symmetric random walks on Z^d**Read:**Sections 4.1 and 4.2 in the file Materials on Random Walks.**Monday, 04/04:**recurrence and transience, simple symmetric random walk on Z^d**Read:**Section 4.2 in the file Materials on Random Walks.**Wednesday, 04/06:**recurrence and transience of general random walks in R^d.**Read:**Section 4.2 in the file Materials on Random Walks.**Friday, 04/08:**recurrence and transience of general random walks in R^d (cont)**Read:**Section 4.2 in the file Materials on Random Walks.**Monday, 04/11:**Trul3 3d random walks are transient. conditional expectation**Read:**Section in the file Materials on Random Walks. Section 4.1 in the book.**Wednesday, 04/13:**Examples and properties of conditional expectations**Read:**Sections 4.1.1 and 4.1.2**Friday, 04/15:**Martingales, submartingales, supermartingales, examples, properties**Read:**Section 4.2**Monday, 04/18:**upcrossing inequality, martingale convergence theorem**Read:**Section 4.2**Wednesday, 04/20:**bounded increments, Doob's decomposition, branching processes**Read:**Sections 4.3.1 and 4.3.4.**Friday, 04/22:**branchung processes, Doob's inequality, L^p maximal inequality**Read:**Sections 4.3.4 and 4.4**Monday, 04/25:**, L^p convergence theorem, applications to branching processes, uniform integrability and L^1 convergence**Read:**Sections 4.4 and 4.6**Wednesday, 04/27:**L^1 convergenc of martingales, dominated convergence theorem for conditional expectations**Read:**Section 4.6**Friday, 04/29:**backwards martingale, application to strong law of large numbers, optional stopping theorem**Read:**Sections 4.7 and 4.8**Monday, 05/02:**optional stopping theorem**Read:**Section 4.8**Wednesday, 05/04:****Read:**Section