### Syllabus

### Lecture Notes for 03/24

### Lecture Notes for 03/26

### Lecture Notes for 03/31

### Lecture Notes for 04/02

### Lecture Notes for 04/07

### Lecture Notes for 04/09

### Lecture Notes for 04/14

### Lecture Notes for 04/16

### Lecture Notes for 04/21

### Lecture Notes for 04/23

### Lecture Notes for 04/28

### Lecture Notes for 04/30

### 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)

### Some Remarks on Homework 7

### Homework 8:5.1.6 (4.1.5 in 5th Ed), 5.1.7( For those of you who do not have the 4th Ed, here is the scanned pdf file), 5.1.9 (4.1.7 in 5th Ed), 5.1.10 (4.1.8 in 5th Ed), due 04/16

### Homework 9: 5.2.4 (4.2.5 in 5th Ed), 5.2.13 (4.2.9 in 5th Ed), 5.2.14 (5.2.10 in 5th Ed), 5.3.1( For those of you who do not have the 4th Ed, here is the scanned pdf file) due 04/23

### Homework 10: 5.3.12 (4.3.11 in 5th Ed), 5.4.4 (4.4.6 in 5th Ed), 5.4.5 (4.4.7 oin 5th Ed), 5.4.6 (4.4.4 in 5th Ed), due 04/30

### Final due 05/06

### 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:**