Instructor: | Ben Steinhurst | ||
Email: | steinhurst at math dot cornell dot edu. (When you send an email message, please identify yourself at the end.) | ||
Office hours: | MW 1:30-2:30 | ||
TA | Pengsheng Ji |
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Email: | pj54 at cornell dot edu | ||
Office Hours: | TBA | ||
Course info: |
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Course description: | This is the first semester of a year-long introduction to probability theory. This semester will focus on basic definitions and results such as the strong law of large numbers, central limit theorem and weak convergence, and discrete time martingales. If time permits we may explore other topics such as random walks and continuous time processes or martingales. A recent knowledge of measure theory will be assumed, but we will briefly review the necessary facts at the beginning of the course. | ||
Course evaluation: | Will be by written homework assigned every two and by a final presentation or exam. Written homework will not be accepted late without prior approval. Feel free to discuss and work on the homework with your classmates but you must deliver your own typed or neatly handwritten version to recieve credit. |
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Course documents: |
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This list will mostly be updated after the fact as a reference for what we have discussed and to post homework assignments.
Lecture | Date | Topics Discussed | See |
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1 | 24 August | Introduction and measure theory review. Probability spaces and extension of measures from semialgebras and algebras to sigma algebras. | Durrett: Ch 1, Rosenthal Ch 1 |
2 | 26 August | More measure theory. Random variables | Durrett: Ch 1 and Appendix A |
3 | 29 August | Yet more measure theory. Integration and properties of expected value. | Durrett: Ch 1 |
4 | 31 August | You thought we were done with measure theory. Think again. Inequalities and change of variables. | Durrett: Ch 1 |
5 | 2 September | Finally done with measure theory. Change of variables and Fubini's theorem. Calculating expected values. | Durrett: Ch 1 HW1 Due 9 Sept 2011. |
5 September | Labor Day. Do some yard work. | ||
6 | 7 September | Definitions of independence | Durrett: Ch 2 |
7 | 9 September | Consequences of independence and calculations | Durrett: Ch 2.1 |
8 | 12 September | Calculations of densities for combinations of independent random variables. | Durrett: Ch 2.1 |
9 | 14 September | Weak Law of Large Numbers, Triangular arrays. | Durrett: Ch 2.2 |
10 | 16 September | Borel Cantelli Lemmas | Durrett: Ch 2.3 HW1 Due 23 Sept 2011. |
11 | 19 September | Second Borel Cantelli Lemma | Durrett: Ch 2.3 |
12 | 21 September | Application of BC Lemma and beginning of Strong Law of Large Numbers | Durrett: Ch 2.3 and 2.4 |
13 | 23 September | Proof of Strong Law of Large Numbers | Durrett: Ch 2.4 |
14 | 26 September | Application of SLLW to Empirical Distributions and beginings of CLT | Durrett: Ch 2.4 and 3.1 |
15 | 28 September | Weak convergence | Durrett Ch 3.2 |
16 | 30 September | Weak convergence | Durrett Ch 3.2 HW 3 is Due 7 October 2011. |
17 | 3 October | Definition of Characteristic functions | Durrett Ch 3.3 |
18 | 5 October | Properties of Characteristic functions, Continuity Theorem | Durrett Ch 3.3 |
19 | 7 October | Proof of Continuity Theorem and examples | Durrett Ch 3.3 |
10 October | Fall break. Look at some leaves. | ||
20 | 12 October | Polya's Criterion Central Limit Theorem | Durrett Ch 3.3 and 3.4.1 |
21 | 14 October | Applications of CLT and Lindeberg-Feller Thm | Durrett 3.4.1 and 3.4.2. HW 4 is Due 21 October 2011. |
22 | 17 October | Proof of Lindeberg-Feller Theorem. Comments about Brownian Motion | Durrett 3.4.2 |
23 | 19 October | Second Proof of Lindeberg-Feller and Erdos-Kac Theorem | Durrett 3.4.3 |
24 | 21 October | Proof of Erdos-Kac Theorem and Poisson Convergence Theorem | Durrett Ch 3.4.3 and 3.6 |
25 | 24 October | Examples and second proof of Poisson Convergence Theorem. Generalized Poisson Convergence Theorem | Durrett: |
26 | 26 October | Poisson Processes and Definition of Levy Processes, Compound, Compensated Poisson Processes | Durrett: 3.6 and class notes. |
27 | 28 October | Guest Lecturer: Nate Eldridge will talk about alpha-stable laws | Durrett Ch 3.7. HW 5 is Due 11 November 2011. |
28 | 31 October | Guest Lecturer: Nate Eldridge will talk abut alpha-stable laws. | Durrett Ch 3.7 |
29 | 2 November | Conditional Probabilities and Filtrations of sigma-algebras in discrete and continuous time. | Durrett Ch 5.1 and class notes. |
30 | 4 November | ||
31 | 7 November | Properties of Stopping times and definitions of martingales. | Class notes. !!!!!!!! Correction for HW 5, X_n,m should have variance 1/n _not_ 1/nt. |
32 | 9 November | Previsible processes and Doob's Stopping in discrete time. | Revuz and Yor II.1 |
33 | 11 November | Lp inequalities in Discrete time | Revuz and Yor II.1 |
34 | 14 November | Lp inequalities in continuous time. | Revus and Yor II.1 |
35 | 16 November | Downcrossing lemma and convergence of martingales in discrete time. | Revuz and Yor II.2 |
36 | 18 November | Cadlag martingales and integrability conditions | Revuz and Yor II.3 |
37 | 21 November | Branching processes | Durrett Ch 5.3 |
38 | 23 November | Discussion | HW 6 is Due 9 December 2011. |
25 November | Day after Thanksgiving. | ||
39 | 28 November | Random walks | Durrett Ch 4.1 |
40 | 30 November | Random walks | Durrett Ch 4.1 |
41 | 2 December | Transience and recurrence | Durrett 4.2 |
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Lasted Updated: 9 December 2011.