Probability Theory I
Fall 2011

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
Email: pj54 at cornell dot edu
Office Hours: TBA
Course info:
Lecture MWF 9:05-9:55
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.
Course documents:

Lecture List with assignments

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
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.