### Syllabus

### Homework 1: 1.6.2, 1.7.3, 2.1.4, 2.1.12 due 02/10

### Homework 2: 2.2.1, 2.2.2, 2.2.4, 2.2.5, due 02/17

### Homework 3: 2.3.2, 2.3.18, 2.4.2, 2.4.3, due 02/24

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

### Homework 5: 3.2.6, 3.2.12, 3.2.13, 3.2.14, due 03/17

### Homework 6: 3.3.16, 3.4.5, 3.4.6, 3.4.12, due 03/31

### Homework 7: 4.1.8, 4.1.9, 4.1.10, 4.1.11, due 04/07

### Homework 8: 5.1.6, 5.1.7, 5.1.9, 5.1.10, due 04/17

### Homework 9: 5.2.4, 5.2.13, 5.2.14, 5.3.1, due 04/24

### Homework 10: 5.3.12, 5.4.4, 5.4.5, 5.4.6, due 04/28

### Final due 05/05

### Class Diary

**Wedensday, 01/18:**probability spaces**Read:**Sections 1.1**Friday, 01/20:**distributions**Read:**Section 1.2**Monday, 01/23:**random variables, integration**Read:**Sections 1.3 and 1.4**Wedensday, 01/25:**integration**Read:**Section 1.4**Friday, 01/27:**properties of the integral**Read:**Section 1.5**Monday, 01/30:**Expected values**Read:**Section 1.6**Wedensday, 02/01:**pi-lambda theorem, product spaces, Fubini's theorem, independence**Read:**Sections 1.7 and 2.1**Friday, 02/03:**independence, distributions, expectations and sums of independent random variables**Read:**Section 2.1**Monday, 02/06:**Kolmogorov's extension theorem, weak law of large numbers**Read:**Section 2.2**Wedensday, 02/08:**weak law of large numbers**Read:**Section 2.2**Friday, 02/10:**Borel-Cantelli lemmas**Read:**Section 2.3**Friday, 02/17:**Borel-Cantelli lemmas (continued), strong law of large numbers**Read:**Sections 2.3 and 2.4**Monday, 02/20:**strong law of large numbers**Read:**Section 2.4**Wedensday, 02/22:**convergence of random series**Read:**Section 2.5**Friday, 02/24:**rates of convergence**Read:**Section 2.5**Monday, 02/27:**weak convergence**Read:**Section 3.2**Wedensday, 03/01:**characterization of weak convergence, Helly's selection theorem, tightness**Read:**Section 3.2**Friday, 03/03:**characteristic functions, the inversion theorem**Read:**Section 3.3**Monday, 03/06:**continuity theorem, moments and derivatives**Read:**Section 3.3**Wedensday, 03/08:**central limit theorem, Lindeberg-Feller theorem**Read:**Section 3.4**Friday, 03/10:**applications of the Lindeberg-Feller theorem, rates of convergence**Read:**Section 3.4**Monday, 03/13:**Test**Wedensday, 03/15:**exchangeable sigma field, Hewitt-Savage 0-1 law, random walks**Read:**Section 4.1**Friday, 03/17:**stopping times, iterates of stopping times, applications to random walks**Read:**Section 4.1**Moday, 03/27:**Wald's equation, simple symmetric random walks**Read:**Section 4.1**Wedensday, 03/29:**Wald's second equations, applications to random walks**Read:**Section 4.1**Friday, 03/31:**recurrence, transience, simple symmetric random walks**Read:**Section 4.2**Monday, 04/03:**recurrence and transience of general random walks**Read:**Section 4.2**Wedensday, 04/05:**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**Monday, 05/01:****Read:****Wedensday, 05/03:**