Probability theory is now considered an integral part of mathematics. The mathematics programmes of most Indian Universities, however, do not include probability theory in much detail. The aim of this school is to give a comprehensive training in probability and stochastic processes to students of the undergraduate and postgraduate programmes. Besides, giving them a chance to interact with researchers in these topics is also a goal of this instructional school.

In addition, over the two weekends, introductory lectures on research topics will be given, which will be followed by interactions with the students.

We propose to run three courses:

- Basic probability theory
- Measure free Stochastic Processes
- Modules of Linear Algebra and Real Analysis

In addition, over the two weekends, introductory lectures on research topics will be given, which will be followed by interactions with the students.

• SPONSORS & CONTACTS •

To attend the school, please fill up the Google form.

Last date of application was: May 14, 2023 (Sunday).

*Application process has been closed!*

NISER Bhubaneswar is located at Jatni, which is around 30 km from **Bhubaneswar City/Railway station (BBS)**. The nearest railway station is: **Khurda Road Junction (KUR)**, which is around 4 km from NISER. One can avail auto-rickshaw service or state run road transportation buses from both the train stations (towards Khurda) to reach NISER. The nearest airport is: **Biju Patnaik International Airport (BBI)**, which is around 28 km from NISER. One can avail *Ola* or pre-paid taxi service from the airport to reach NISER. See Map

Any EIGHT of the following speakers will deliver talks according to their availability and mutual understanding.

IACS, Koklata

ISI, Delhi

ISI, Delhi

ISI, Kolkata

ISI, Kolkata

CMI, Chennai

NISER, Bhubaneswar

NISER, Bhubaneswar

Ashoka University, Delhi

IISER, Pune

NISER, Bhubaneswar

IIM, Bengaluru

ISI, Kolkata

**Basic Probability Theory:** Orientation, Elementary concepts: experiments, outcomes, sample space, events. Discrete sample spaces and probability models. Combinatorial probability and urn models; Conditional probability and independence; Random variables – discrete and continuous; Expectations, variance and moments of random variables; Markov inequality, Chebycheff inequality, First and second moment method (in tutorial with applications), Transformations of univariate random variables; Jointly distributed random variables; joint density and change of variables formula, independence and conditional distribution; probability transforms (if time permits), Modes of convergence, Borel-Cantelli lemma, WLLN under 2nd moment assumption and SLLN under 4th moment assumption, CLT (statement with applications)

**Stochastic Processes:** Simple symmetric Random Walk – reflection princliple, return to origin, Arc-sine law, Discrete Markov chains with countable state space, example of two state Markov chain, Recurrence, transience, criterion for recurrence, positive and null recurrence; Irreducibility and decomposition of state space, class property, stationary distributions, ratio limit theorem, periodicity, limit theorems, reversible chains. Generating functions; Several illustrations including the Gambler's ruin problem, queuing chains, birth and death chains etc., Branching process, Poisson process, continuous time Markov chain with countable state space, continuous time birth and death chains.

**Analysis:** Sequences, limits, limsup and liminf; infinte series, convergence and absolute convergence, Limit of a function at a point, continuity, intermediate value theorem, differentiation, Rolle's theorem, Taylor's theorem Power series – radius of convergence, term by term differentiation and integration, Abel's theorem, Cauchy-Schwarz Inequality for infinite series and integrals.

**Linear Algebra:** Matrices, Matrix Operations, Determinants, Matrix Representation of system of linear Equations, Vector Spaces and Subspaces, Basis of a Vector Space, Linear transformations, matrix representation, Inner product spaces and orthogonality, Cauchy-Schwarz Inequality for finite sum Eigenvalues, Classification and Transformation of Quadratic Forms, Diagonalisation of real symmetric matrices.

**References:**

- A. Ramachandra Rao and P. Bhimasankaram: Linear Algebra.
- G. F. Simmons: Introduction to Topology and Modern Analysis
- S. M. Ross: A first course in Probability
- Jacod & Protter: Probability Essentials
- W. Feller: Introduction to the Theory of Probability and its Applications, Vol. 1.
- P.G. Hoel, S.C. Port and C.J. Stone: Introduction to Stochastic Processes.
- (Lecture notes from the speakers, if available)

Accommodation will be provided to all the participants in NISER hostels during the period of the school. **Participants may plan to reach NISER preferably on Sunday, June 25, 2023**. The name of the hostel allotted to the participants will be notified here.

**Message for the shortlisted candidates:**

Your candidature for the Basic Instructional School on Stochastic Processes has been shortlisted. Kindly confirm your participation by sending the copy of travel documents by **21st May 2023 6AM** to **aissp@niser.ac.in** . Please note the following:

- Sleeper class train fare will be reimbursed to all the outstationed participants.
- The competent authority of NISER has approved free hostel accommodation from June 25, 2023 to July 22, 2023 for participants. Participants may bring their bed cover and other personal use materials.
- Outstationed participants will be provided ₹260/- per day for their daily food expenses subject to the attendance of the classes. Participants may avail their food from the NISER hostels. Due amount will be credited to their bank accounts in the last week of the program.

• Aakriti | Deen Dayal Upadhyaya College |

• Abhinandan Mohapatra | Fakir Mohan Autonomous College |

• Aindrila Rakshit | Lady Brabourne College |

• Akash Kumar Heera | Visva-Bharati, |

• Alisha Rani Sahu | Larambha College |

• Animesh Baral | Visva-Bharati |

• Anirban Nandy | Hooghly Mohsin College |

• Arif Ali | Ramakrishna Mission |

• Arohini Ghosh | Bethune College |

• Ashutosh Mahanta | Fakir Mohan Autonomous College |

• Avik Shakhari | Institute of Mathematics and Applications |

• Devansh Mishra | Kisan P G College |

• Dishari Bhaduri | Madras Christian College |

• Ipsita Panda | Veer Surendra Sai University of Technology |

• Isha Gupta | Deen Dayal Upadhyaya College |

• Ishita Mandal | Hooghly Mohsin College |

• Lekhraj | Kisan Post Graduate College |

• Lopamudra Barad | Banki (Autonomous) College |

• Nidhi Kumari | Banaras Hindu University |

• Parmita Das | Deen Dayal Upadhyaya College |

• Prajna Paramita Pradhan | Utakl University |

• Prateek Kumar | KIIT University |

• Priyanka | Banasthali Vidyapith |

• Priyanka Vishnudas Ghuge | NIT Rourkela |

• Rajdeep Pathak | Jadavpur University |

• Raktim Ray | Jadavpur University |

• Sakthipriya V | Seethalakshmi Ramaswami College |

• Samaroha Chatterjee | Institute of Mathematics and Applications |

• Sarbani Pal | Bethune College |

• Shubham Dahiya | Deen Dayal Upadhyaya College |

• Sonu Verma | South Asian University |

• Swati Pradhan | Utkal University |

• Thaneshwar Prasad Choudhary | Mohanlal Sukhadia University |

• Vishnu Unnikrishnan | St. Thomas College (Autonomous) |

• Yash Gupta | Raj Rishi Govt. Autonomous College |

Wait-listed participants

• Abhinandan Mohanty | National Institute of Technology, Rourkela |

• Alok Naik | Veer Surender Sai University of Technology |

• Anandi Roy | IISER-Berhampur |

• Gaurav Kumar | Indian Statistical Institute |

• Mayank Jangid | Indian Statistical Institute |

• R Vedanta | Indian Statistical Institute |

• Rishith Reddy V | IISER Tirupati |

- will be updated later -

Postal contact address:

Dr. Nabin Kumar Jana (Readder-F) | Dr. Kaushik Majumder (Asst. Professor)Organisers, BISSP 2023

School of Mathematical Sciences

National Institute of Science Education and Research, Bhubaneswar

PO: NISER, Via: Jatni, District: Khurda, Odisha - 752050

- +91-674-2494090 / 2494109
- nabinjana | kaushikmajumder [@niser.ac.in]

- School of Mathematical Sciences, NISER

- +91-674-2494090 / 2494109

- aissp@niser.ac.in