+91-674-249-4082

Submitted by nabinjana on 8 March, 2018 - 11:57

Date/Time:

Monday, June 25, 2018 (All day) to Saturday, July 21, 2018 (All day)

Advanced instructional school on Stochastic Processes will be organized at NISER, Bhubaneswar sponsored by National Centre for Mathematics and Indian Academy of Sciences during 25th June to 21st July 2018.

Aims and Objectives: Most Indian Universities do not have a rigorous study on probability. The aim of this school is to give a comprehensive training to students in a master’s degree/PhD programme on probability and stochastic processes. Also to give them a chance to interact with researchers

in these topics.

We propose to run three courses in parallel

- Analysis,
- Measure-theoretic probability
- An introductory course on stochastic processes.

There will be around 36 hours of lectures including tutorials per topic over a 4 week period. In addition, over two weekends we plan to invite active researchers in probability to present introductory lectures on a research topic and interact with students.

Students of Master/Ph.D. in Mathematics, Physics, Statistics, Computer Science and Electrical Engineering;

B.Tech./M.Tech. (in CS and EE) and M.Phill (in Physics, Mathematics, Statistics)

Application can be made through either of the following websites on/before **01st May 2018.**

Around 40 students will be selected from the applications received and chosen on the basis of the student's background, educational qualifications and an **assessment as to the extent to which the candidate would benefit from the programme. **

Travel, local conveniences and course materials will be provided to the participants.

Speakers & Tutors:

(a) Anish Sarkar,(AS) ISI, Delhi

(a) Anish Sarkar,(AS) ISI, Delhi

(b) Antar Bandyapadhyay, (AB) ISI, Delhi

(c) Arijit Chakrabarty,(AC) ISI, Kolkata

(d) Krishanu Maulik, (KM) ISI, Kolkata

(e) Nabin Kumar Jana, (NKJ) NISER, Bhubaneswar

(f) Parthanil Roy, (PR) ISI, Bangalore

(g) Rajat Subhra Hazra,(RSH) ISI, Kolkata

(h) Rahul Roy,(RR) ISI, Delhi

- Nabin Kumar Jana, NISER, Bhubaneswar
- Rahul Roy, ISI, Delhi

`aissp 'at' niser.ac.in`

Syllabus:

We plan to cover the following topics in this AIS. The proposal is a bit ambitious and we will tune it according to the level of students. Our aim is also to have a follow-up workshop later where selected students from this group will be introduced to more advanced topics.

- Analysis:Metric spaces, open/closed sets, sequences, compactness, completeness, continuous functions and homeomorphisms, connectedness, product spaces, Baire category theorem, completeness of C[0, 1] and L p spaces, Arzela-Ascoli theorem Analytic functions, Cauchy-Riemann equations, polynomials, exponential and trigonometric functions Contour integration, Power series representation of analytic functions, Liouville’s theorem,Cauchy integral formula, Cauchy’s theorem, Morera’s theorem, Cauchy-Goursat theorem Singularities, Laurent Series expansion, Cauchy residue formula, residue calculus. Meromorphic functions, Rouche’s theorem.Fractional linear transformations.

Reference Texts:- G. F. Simmons: Introduction to Topology and Modern Analysis
- J. C. Burkill and H. Burkill: A second course in mathematical Analysis
- J. Conway: Functions of one complex variable
- L. Ahlfors: Complex Analysis

- Measure Theoretic Probability:
- Motivation: Doing integration beyond Riemann theory, infinite tosses of a fair coin

Fields, sigma-fields, measures, sigma-finite/finite/probability measures, properties, statement of Caratheodory extension theorem (outline of idea, if time permits). Monotone class theorem, Dynkin’s pi-lambda theorem. Radon measures on finite-dimensional Borel sigma-field, distribution functions, correspondence between probability measures on Borel sigma-field and probability distribution functions.Measurable functions, basic properties, sigma-fields generated by functions, integration of measurable functions, properties of integrals, MCT, Fatou’s Lemma, DCT, Scheffes theorem.Chebyshev’s, Holder’s and Minkowski’s inequalities. L p spaces. Finite product of measurable spaces, construction of product measures, Fubinis theorem. Probability spaces, random variables and random vectors, expected value and its properties. Independence.Various modes of convergence and their relation. Uniform integrability (if time permits). The BorelCantelli lemmas. Weak Law of large numbers for i.i.d. finite mean case. Kolmogorov 0-1 law, Kolmogorov’s maximal inequality. Statement of Kolmogorov’s three-Series theorem (proof if time permits). Strong law of large numbers for i.i.d. case.

Characteristic functions and its basic properties, inversion formula, Levy’s continuity theorem.Lindeberg CLT, CLT for i.i.d. finite variance case, Lyapunov CLT. - Reference Texts:

1. Probability and Measure Theory: Robert B. Ash & Catherine A. Doleans-Dade

2. A Course in Probability Theory: Kai Lai Chung

3. Probability and Measure: Patrick Billingsley

4. Probability Theory: Y. S. Chow and H. Teicher

5. Probability: Theory and Examples: Rick Durrett

- Motivation: Doing integration beyond Riemann theory, infinite tosses of a fair coin
- Introduction to Stochastic Processes:
- Discrete Markov chains with countable state space, Examples including 2-state chain, random walk, birth and death chain, renewal chain, Ehrenfest chain, card shuffling, etc.Classification of states, recurrence and transience; absorbing states, irreducibility, decomposition of state space into irreducible classes, Examples.

Absorbing chains, absorption probabilities and mean absorption time, fundamental matrix Stationary distributions, limit theorems, positive and null recurrence, ratio limit theorem, reversible chains. Periodicity, cyclic decomposition of a periodic chain, limit theorems for aperiodic irreducible chains. Introduction to MCMC, perfect sampling Review of Poisson process and its properties, non-homogeneous and compound Poisson processes, Simple birth and death processes, a brief introduction to general continuous time Markov chains, Kolmogorov equations

Reference Texts

1. W. Feller: Introduction to the Theory of Probability and its Applications, Vol. 1.

2. P. G. Hoel, S. C. Port and C. J. Stone: Introduction to Stochastic Processes.

3. J. G. Kemeny, J. L. Snell and A. W. Knapp: Finite Markov Chains.

- Discrete Markov chains with countable state space, Examples including 2-state chain, random walk, birth and death chain, renewal chain, Ehrenfest chain, card shuffling, etc.Classification of states, recurrence and transience; absorbing states, irreducibility, decomposition of state space into irreducible classes, Examples.

Tentative Time-Table

Week 1 | |||

09:00-11:00 | 11:30- 12:30 | 14:30-16:30 | 17:00-18:00 |

Analysis I | Tutorial | Introduction to Stochastic Processes I | Tutorial |

Week 2 | |||

Analysis II | Tutorial | Measure theoretic probability I | Tutorial |

Week 3 | |||

Introduction to Stochastic Processes II | Tutorial | Measure theoretic probability II | Tutorial |

Week 4 | |||

Introduction to Stochastic Processes III | Tutorial | Measure theoretic probability III | Tutorial |

To be uploaded around 7th May 2018.

**School of Mathematical Sciences**

NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050

Tel: +91-674-249-4081

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