### Syllabus

### Lecture Notes for 03/24

### Lecture Notes for 03/26

### Homework 1: 1.6.2, 1.7.3, 2.1.4 (2.1.1 in 5th Ed.), 2.1.12 (2.1.9 in 5th Ed), due 02/06

### Homework 2: 2.2.1, 2.2.2, 2.2.4, 2.2.5, due 02/18

### Homework 3: 2.3.2,(2.3.18 in 5th Ed.) 2.3.18.(2.3.15. in 5th Ed.) 2.4.2.(2.4.1. in 5th Ed) 2.4.3.(2.4.2. in 5th Ed.) due 02/27

### Homework 4: 2.5.1, 2.5.2, 2.5.9, 2.5.10 due 03/05

### Homework 5: 3.2.6, 3.2.12, 3.2.13, 3.2.14 due 03/24

### Homework 6: 3.3.16, (3.3.13 in 5th Ed.) 3.4.5, 3.4.6, 3.4.12, due 04/02

### Homework 7: 4.1.8, 4.1.9, 4.1.10, 4.1.11, due 04/09 ( For those of you who do not have the 4th Ed, here is the scanned pdf file)

### Homework 8: due

### Homework 9: due

### Homework 10: due

### Final due 05/05

### Class Diary

**Tuesday, 01/21:**probability spaces, distributions**Read:**Sections 1.1 and 1.2**Thursday, 01/23:**random variables, integration**Read:**Sections 1.3 and 1.4**Tuesday, 01/28:**Jensen's inequality, Holder's inequality, bounded convergence theorem, Fatou's lemma, monoytone convergece theorem, dominated convergence theorem, expectation, Chebyshev's inequality**Read:**Sections 1.5, 1.6.1, 1.6.2**Thursday, 01/30:**Change of variable formula, product measure, pi-system, lambda-system, pi-lambda theorem, Fubini's theorem, independence**Read:**Sections 1.6.3, 1.7, 2.1.1**Tuesday, 02/04:**Sums of independent random variables, constructing independent random varaibles, Kolmogorov's extension theorem, L^2 weak law**Read:**Sections 2.1.2, 2.1.3, 2.1.4, 2.2.1**Thursday, 02/06:**weak law for triangular arrays, coupon collector's problem, truncation, weak law of large numbers, St. Petersburg paradox, Borel-Cantelli Lemma**Read:**Sections 2.2.2, 2.2.3, 2.3**Tuesday, 02/11:**weak law with finite second moment, second Borel-Cantellui lemma, example on head runs**Read:**Sections 2.3**Thursday, 02/13:**strong law of large numbers, renewal theory, Glivanko-Cantelli theorem, tail sigma field, Kolmogorov's 0-1 law, Kolmogorov's maximal inequality**Read:**Section 2.4, 2.5**Tuesday, 02/18:**Kolmogorov's 3 series theorem, strong law of large numbers, rates of convergence, infinite mean**Read:**Sections 2.5, 2.5.1, 2.5.2**Thursday, 02/20:**weak convergence, characterization of weak convergence, Helly's selection theorem, tightness**Read:**Section 3.2**Tuesday, 02/25:**characteristic functions, inversion formula, contionuity theorem**Read:**Section 3.3.2, 3.3.2**Thursday, 02/27:**moments and derivatives, central limit theorem, Lindeberg-Feller theorem**Read:**Sections 3.3.3, 3.4.1, 3.4.2**Tuesday, 03/03:**applications of the Lindeberg-Feller theorem, rates of convergence, exchangeable sigma field,**Read:**Section 3.4.2, 3.4.4, 4.1**Thursday, 03/05:**Hewitt-Savage 0-1 law, stopping times, iterates of stopping times, application to random walks**Read:**Section 4.1**Tuesday, 03/10:**Wald's equation, simple symmetric random walks, Wald's second equation, applications to random walks**Read:**Section 4.1**Thursday, 03/12:**test. Will cover materials from Chapter 2.**Read:**Section**Tuesday, 03/24:****Read:**Section**Thursday, 03/26:****Read:**Section**Tuesday, 03/31:****Read:**Section**Thursday, 04/02:****Read:**Section**Tuesday, 04/07:****Read:****Thursday, 04/09:****Wedensday, 04/14:****Read:****Friday, 04/16:****Read:**Section**Moday, 04/21:****Read:**Section**Wedensday, 04/23:****Read:**Section**Friday, 04/28:**recurrence, transience, simple symmetric random walks**Read:**Section 4.2**Monday, 04/30:**recurrence and transience of general random walks**Read:**Section 4.2**Wedensday, 05/05:**recurrence criterion for general random walks**Read:**Section 4.2**Friday, 04/07:**conditional expectation**Read:**Section 5.1**Monday, 04/10:**properties of conditional expectations, martingales**Read:**Sections 5.1 and 5.2**Wedensday, 04/12:**properties of (sub)martingales, upcrossing inequality**Read:**Section 5.2**Friday, 04/14:**martingale convergence theorem, Doob's decomposition, martingales with bounded increments**Read:**Sections 5.2 and 5.3.1**Monday, 04/17:**Radon-Nikodym derivatives, branching processes,**Read:**Section 5.3.3**Wedensday, 04/19:**Doob's maximal inequality, L^p maximal inequality**Read:**Section 5.4**Friday, 04/21:**L^p convergence theorem, L^2 martingales**Read:**Section 5.4**Monday, 04/24:**uniform covergence, convergence in L^1**Read:**Section 5.5**Wedensday, 04/26:**backwards martingale**Read:**Section 5.6**Friday, 04/28:**Optional stopping theorem**Read:**Section 5.7