Venue: Feynman Lecture Hall
Class Timings: Tuesdays and Thursdays from 9:45 AM-11: 15 AM (Jointly offered at ICTS and ISI).
First Meeting: 8 August 2024
Course Syllabus:
- Revision of Measure theory: probability spaces, distributions, random variables, standard random variable examples, expected value, inequalities (Holder, Cauchy-Schwarz, Jensen, Markov, Chebyshev) , convergence notions(convergence in probability and almost sure, Lp), application of DCT, MCT, Fatou with examples, Revision of Fubinis theorem.
- Independence, sum of random variables, constructing independent random vari ables, weak law of large numbers, Borel-Cantelli lemmas, First and Second Moment methods, Chernoff bounds and some applications.
- Strong law of large number, Kolmogorov 0-1 law. Convergence of random series. Kolmogorovs three series theorem.
- Weak convergence, tightness, characteristic functions with examples, Central limit theorem (iid sequence and triangular array).
- Discrete Time Markov Chains
References:
- 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
Other References
- Rick Durrett: Probability (Theory and Examples).
- Patrick Billingsley: Probability and measure.
- Robert Ash: Basic probability theory.
- Leo Breiman: Probability.
- David Williams: Probability with Martingales.
Prerequisite: Undergraduate Probability course and Undergraduate Advanced Calculus/Real Analysis course. Measure Theory course will be helpful but not essential.
- Teacher: Siva Athreya
Credit Score: 4