Introduction to Stochastic Processes

Basic Information

• Instructor : Siva Athreya .
• Email : athreya@isibang.ac.in or athreya@icts.res.in
• Scoring : Final Exam: 50%, Midterm Exam: 25%, Homework: 25%.

Syllabus:

F rom B.Math. (Hons.) III Year students-Brochure

• DISCRETE-TIME MARKOV CHAINS: Classification of states, Stationary distribution, reversibility and convergence. Random walks and electr ical networks. Collision and recurrence.
• DISCRETE-TIME MARTINGALES: Optional Stopping theorem, Martingale convergence theorem, Doob’s inequality and convergence.
• BRANCHING PROCESSES: Model definition. Connection with martingales. Probability of survival. Mean and variance of number of individuals.
• BASIC PROBABILISTIC INEQUALITIES AND APPLICATIONS: First and Second Moment methods. Applications to Longest increasing subsequences, Random k-Sat problem and connectivity threshold for Erdos-Renyi graphs. Chernoff bounds and Johnson-Lindenstrauss lemma.

References and Texts

• [ADS]Chapters 1-6Probability and Statistics with Examples using R
• [LHdHGVZ] Random Walks by Luca Avena, Markus Heydenreich, Frank den Hollander, Evgeny Verbitskiy, Willem van Zuijlen
• [AF]Reversible Markov Chains and Random Walks on Graphs by David Aldous and Jim Fill.
• [AS] Chapter 6 in Measure and Probability by Siva Athreya and V.S. Sunder.
• [D] Probability: Theory and Examples by Rick Durrett.
• [G] Discrete Stochastic Processes by Robert Gallager.
• [LPW]Markov Chains and Mixing Times by David A. Levin, Yuval Peres, and Elizabeth L. Wilmer.
• [N] Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) by James Norris.

Classes:

Day and Time : Wednesday : 2:00-3:00pm, 3:10pm-4:10pm, 4:10pm-5:10pm, 5:10pm-6:10pm at ISI Bangalore Centre


Homework:

There will be regular Homework assignments given during the course. The problems given there will be a small selection so please do try other problems for practice from the texts. You are encouraged to work together on solving the problems but please write up solutions without consulting anyone else. All Homeworks should be either uploaded onto the respective Moodle platform by 6:30pm on the due date or as mentioned in class.