NCMW - Discrete stochastic models (2025)

Speakers and Syllabus

Name of the Speakers with their affiliation.

No. of Lectures

Detailed Syllabus

Moumanti Podder
IISER Pune

4
(each 1.5 hrs)

Introduction to discrete and continuous time Markov chains and their various applications, random walks (including simple symmetric random walks on the infinite d-dimensional integer lattice), irreducibility and equivalence classes, transience and recurrence, branching processes (with emphasis on Galton-Watson branching
processes) and their applications..

Anup Biswas
IISER Pune

5
(each 1.5 hrs)

 Conditional expectation, definition, existence and uniqueness, properties of conditional expectation, relation with the classical conditional probability, conditional inequalities and limit theorems, conditional Bayes’ formula, regular conditional probability

Anish Sarkar
ISI Delhi

5
(each1.5 hrs)

 Martingales: Definitions & Examples, Submartingales, Functions of Martingales, Inequalities and some applications, Convergence Theorems, Reversed Martingales, Central Limit Theorem (time permitting).

Krishanu Maullik
ISI Kolkata

5
(each 1.5 hrs)

Heavy Tailed Distributions

Definitions, properties and examples of heavy-tailed, subexponential and regularly varying distributions, Karamata properties of regular variation, one large jump principle for subexponential distributions, tails of random sums, generalized central limit theorem, maximum of random walk with negative drift, degree distribution of preferential attachment graphs, record processes.

Rahul Roy
ISI Delhi

5
(each 1.5 hrs)

Percolation: Percolation on trees, bond percolation on the square lattice, oriented percolation in 2 dimensions. Introduction to percolation theory and why is it different from standard stochastic processes (here the randomness is in the medium and not in the motion), the notion of phase transition, using branching process techniques to explicitly calculate the phase transition point for percolation on trees. Showing that the phase transition point for d-dimensional lattice is non-trivial. Discussing Correlation inequalities (i.e., the FKG and BK inequalities), proving the subadditive lemma. Discussion of the notion of ergodicity, invariant sigma field and state the ergodic theorem. Proving the uniqueness of the unbounded cluster (i.e. the Burton-Keane theorem). Showing that p_c = 1/2 for bond percolation on the square lattice. If time permits, discussion of site percolation on the square lattice.

References:

  1. Patrick Billingsley. Probability and measure. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1995.

  2. Sheldon M. Ross. Stochastic Processes. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1995.

  3. Samuel Karlin, Howard E. Taylor. A First Course in Stochastic Processes. Academic Press, San Diego, CA, 1975.

  4. Krishna B. Athreya and Peter E. Ney. Branching Processes. Springer-Verlag Berlin, Heidelberg, 1972.

  5. William Feller. An Introduction to Probability Theory and its Applications. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1957.

  6. David Williams. Probability with Martingales. Statistical Laboratory, University of Cambridge, 1991.

  7. Ronald Meester, Rahul Roy. Continuum Percolation. Cambridge University Press, 1996.

  8. Geoffrey Grimmett. Percolation. Springer-Verlag Berlin, Heidelberg, 1999.

  9. Paul Embrechts, Claudia Kluppelberg, Thomas Mikosch. Modelling Extremal Events: for Insurance and Finance. Vol 33, Springer Science and Business Media, Berlin, 2013.

  10. Jayakrishnan Nair, Adam Wierman, Bert Zwart. The Fundamentals of Heavy Tails: Properties, Emergence and Identification. Cambridge University Press, 2022.

Names of the tutors with their affiliation:

  1. Dhruv Bhasin, IISER Pune

  2. Archi Roy, IISER Pune

  3. Tamojit Sadhukhan, ISI Kolkata

  4. Aritra Majumdar, ISI Kolkata

 

 

Time Table

 

 

Tentative time-table, mentioning names of the speakers and tutors with their affiliation:

Day

Date

Lecture 1
9.30
to
11.00

Tea
11.05
to
11.25

Lecture 2
11.30
to
1.00

Lunch
1.05
to
1.55

Lecture 3
2.00
to
3.30

Tea
3.35
to
3.55

Discussion
4.00
to
5.00

Snacks
5.05
to
5.35

 

 

(name of the speaker)

 

(name of the speaker)

 

(name of the speaker)

 

(Name of the tutor)

 

Thu

July 10

AB

 

AB

 

MP

 

AB

 

Fri

July 11

AB

 

MP

 

MP

 

MP

 

Sat

July 12

AB

 

AB

 

MP

 

AB

 

Mon

July 14

AS

 

AS

 

KM

 

MA+AS

 

Tues

July 15

KM

 

KM

 

RR

 

KM

 

Wed

July 16

AS

 

KM

 

RR

 

KM

 

Thu

July 17

AS

 

KM

 

RR

 

RR

 

Fri

July 18

AS

 

RR

 

RR

 

MA+AS

 

Full forms for the abbreviations of speakers and tutors:

  • MP: Moumanti Podder
  • AS: Anish Sarkar
  • KM: Krishanu Maulik
  • RR: Rahul Roy
  • AB: Anup Biswas
  • MA: Maruf Alam
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