C-12 Probability Theory (Master of Mathematics)

Basic Information

Instructor : Siva Athreya .
Teaching Assistant : Sayantan Maitra and Aritra Mondal.
Email : athreya@isibang.ac.in
Scoring : Final Exam: 50%, Midterm Exam: 25%, Quizzes: 20%, Homework: 5%.
Syllabus: Course C-12 in Brochure

Texts (with Chapters to be followed)

[J06] Chapters 1-5,9,10,11 in A First Look at Rigorous Probability Theory by Jeffrey S. Rosenthal
[AS08] Chapters 1-5 in Measure and Probability by Siva Athreya and V.S. Sunder.
[CT01]Chapter 3-5 in Probability Theory: Independence, Interchangeability, Martingales (Springer Texts in Statistics) 3rd Edition by Yuan Shih Chow and Henry Teicher

References From Brochure

Rick Durrett: Probability (Theory and Examples).
Patrick Billingsley: Probability and measure.
Robert Ash: Basic probability theory.
Leo Breiman: Probability.
David Williams: Probability with Martingales.

Classes:

Day and Time

August 1st, 4th, 18th 2022.
September 26th, 29th 2022.
October 3rd, 6th, 2022.

Homework and Quizzes:

Week #	Notes in Each Week	Homework (Due Dates)	Topics
1	Week-1	(August 10th, 2022) Homework 1	Probability spaces, distributions. [J06] Sections: 1.2, 2.1-2.6 [AS08]Sections: 1.1-1.6 Homework Solution Quiz 1 Solution
2	Week-2	(August 17th, 2022) Homework 2	Random Variables Independence $\limsup_{n\rightarrow\infty} A_n$, $\liminf_{n\rightarrow\infty} A_n$ . Borel-Cantelli Lemma [J06] Sections: 3.1,3.2,3.4 Homework 2 Solution
3	Week-3	(August 25th, 2022) Homework 3	Kolmogorov 0-1 law. [J06] Sections: 3.5 [AS08]Section:3.2 Introduction to bond Percolation On ${\mathbb Z}^d$ Note on Construction on Independent Random Variables by Manjunath Krishnapur. Solution Quiz 3 Solution
4	Week-4	(September 1st, 2022) Homework 4	Distribution of Random Variables. Expectation of Random Variables. Markov Inequality, Cauchy-Schwarz-Bunyakovsky inequality, Jensen's Inequality. [J06] Sections:4.1,4.2,4.3,4.4, 5.1 [AS08]Section:2.1,2.2,2.3,3.1 Homework 4 Solution Quiz 4 Solution
5	Week-5	(September 8th, 2022) Hw 5	Convergence in Probability, Almost sure Convergence Monotone Convergence Theorem Sequence of Independent Random Variables(Construction), Covariance [J06] Sections: 5.2,5.3 [AS08]Section:4.1,4.2,5.2 Homework 5 Solution Quiz 5 Solution
6	Week-6	(September 15th, 2022) Hw 6	First moment method Second moment method (Application: Coupon Collector and Bond Percolation on binary Tree) Chernoff bound Probability Theory notes by Manjunath Krishnapur. Homework 6 Solution Quiz 6 Solution
7	Week-7	Midterm (September 20th, 2022)	Dominated Convergence Theorem Fatou's Lemma Moment Generating Function [J06] Sections: 9.3 [AS08]Section:2.2
8	Week-8	(September 29th, 2022) Worksheet (October 6th, 2022) Homework 8	Fubini's Theorem Large Deviations --brief motivation
9	Week-9	(October 3rd, 2022) Worksheet 1 (October 5th, 2022) Worksheet 2	Cramers Theorem A Short Proof of Cramer’s Theorem in ${\mathbb R}$ by Rapha$\ddot{\mbox{e}}$l Cerf and Pierre Petit
10	Week-10	(October 20th, 2022) Homework 9	Convergence in Probability : Equivalent Formulations. Convergence Almost surely: Equivalent Formualtions. Levy's Theorem and inequalities. [CT01] Sections:3.3 [AS08]Section:5.2 Homework Solution
11	Week-11	(October 27th, 2022) Homework 10	3-Series Theorem Weak Convergence Basics [J06] Sections:10.1 [CT01]Sections:5.1 Homework Solution
12	Week-12	(November 3rd, 2022) Homework 11	Portmanteau Theorem (Equivalent Formulations of Weak Convergence) Characteristic Functions [J06] Sections:10.1,11.1 [AS08]Section:5.1 Homework Solution
13	Week-13	(November 10th, 2022) Homework 12	Skorokhod Theorem Fourier Inversion Continuity Theorem Central Limit Theorem [J06] Sections:11.1, 11.2 [AS08]Section:5.1,5.2,5.3 Homework Solution
14	Week-14	--	Strong Law of Large Numbers Essence of Law of Large Numbers by M. Keane [AS08]Section:5.4

Last Modified: November 17th, 2022.