Asymptotic theory is a central unifying theme to both theoretical and applied statistics. This theory underlies much of the work on different topics such as maximum likelihood estimation, likelihood ratio tests and some of their variants, the bootstrap, etc. Teachers and young research students of mathematical statistics need to have a proper background of the various results related to asymptotic theory of probability and statistics. This IST syllabus intends to train the participants tackle asymptotic problems of statistics in their research. The instructional school will begin with the introduction to basic concepts of convergence of sequence of random variables, their interrelationships, weak/strong laws of large numbers and central limit theorems. This will be followed by classical asymptotic theory of estimation based on independent and dependent sequences. Theory of empirical processes and their applications to various inference problems will be discussed further. Contiguity of probability measures, Le Cam’s local asymptotic normality (LAN) theory in quadratic mean differentiable (QMD) families will be introduced subsequently. Bootstrap methods and their applications will be also discussed.

**Name & Affiliation of Speakers:**

- B. V. Rao, Chennai Mathematical Institute, Chennai
- M. B. Rajarshi, Savitribai Phule Pune University, Pune
- U. V. Naik-Nimbalkar, IISER, Pune
- S. R. Deshmukh, Savitribai Phule Pune University, Pune
- T. V. Ramanathan, Savitribai Phule Pune University, Pune
- V. Swaminathan, Chennai Mathematical Institute, Chennai

**Proposed Guest Speakers: (Any two**)

- J. V. Deshpande, CMI/SPPU
- M. S. Prasad, Bharati Vidyapeeth University, Pune
- Mohan Kale, Department of Statistics, SPPU, Pune

**Name & Affiliation of Course Associates:**

1. Madhuri Kulkarni, Department of Statistics, Savitribai Phule Pune University, Pune

2. Akanksha Kashikar, Department of Statistics, Savitribai Phule Pune University, Pune

3. Anuj Mishra, Department of Statistics, Savitribai Phule Pune University, Pune

4. Sagar Pandhare, Department of Statistics, Savitribai Phule Pune University, Pune

5. Mukund Ramtirthkar, Department of Statistics, Savitribai Phule Pune University,Pune

**Syllabus to be covered in 6 units and corresponding speakers**

Speaker |
Detailed Syllabus |

V. Swaminathan, CMI (VS) (6L + 4 T) | Unit 1: Prerequisites of Probability Theory Indicative subunits: Modes of Convergence of sequence of random variables and their inter relationships, Borel-Cantelli’s Lemmas, Kolmogorov’s Zero-One Law, Slutsky’s theorems, Integrals in probability space, Lp spaces, uniform integrability, Vitali’s theorem, Scheffe’s theorem, Dominated convergence theorems, results on interrelations between uniform integrability, convergence in probability and convergence of moments |

B. V. Rao, CMI, (BVR) (6L + 4 T) | Unit 2: Limit Theorems Indicative subunits: Laws of large numbers, Central Limit Theorems (i.i.d. as well as dependent set up such as martingale, mixing etc.), Conditional expectations and convergence of conditional moments. Weak convergence. |

S. R. Deshmukh, (SRD) (3L + 3 T) | Unit 3: Classical CAN Theory (i.i.d. and dependent set up) Indicative subunits: Consistency (strong, weak and uniform), CAN estimators, Best Asymptotic Normal (BAN) estimators, Asymptotic efficiency, Hodge’s estimator, Delta method, Maximum likelihood estimators (MLE), CAN-ness of MLE in Cramer-regular families, scoring procedures, Inconsistent MLEs (Neyman-Scott problems) |

U. V. Naik- Nimbalkar (UVN)(3L + 3 T) | Unit 4: Empirical Processes: Theory and Applications Indicative subunits: Empirical measures, Empirical process, Glivenko-Cantelli (GC) Lemmas, Weak convergence of empirical processes, Donsker’s Theorem, GC class and Donsker class of empirical functions, Brownian bridge, Uniform CLT and Gaussian processes Asymptotic properties of robust estimators, nonparametric MLE, Rates of convergence, Goodness of fit tests |

T. V. Ramanathan, (TVR) (3L + 3 T) | Unit 5: Local Asymptotic Normality (LAN) Theory Indicative subunits: Contiguity of probability measures, Local Asymptotic Normality (LAN), Efficiency of estimators, Le Cam’s Lemmas, Limitations and redundancy of Cramer’s regularity conditions, A class of non-differentiable location models which yield efficient estimators, Quadratic mean differentiable (QMD) families, Applications of LAN theory of QMD families, |

M. B. Rajarshi, (MBR) (3L + 3 T) | Unit 6: Bootstrap Methods & Asymptotics Indicative subunits: Bootstrap distribution, Delta theorems for bootstrap, Bootstrap confidence intervals, Applications in regression, autoregressive models and related stochastic models, Edgeworth expansions and bootstrap, Block bootstrap, Limitations of bootstrap |

Guest Speaker-1, 2 hrs (during T hrs) | 2 Lectures on proposed theme related topics |

Guest Speaker-2, 2 hrs (During T hrs) | 2 Lectures on proposed theme related topics |

**Reference Books:**

- Basawa, I. V. and Rao, B. L. S. P (1980), Statistical Inference for Stochastic Processes,Academic Press, London
- Billingsley, P. (1986), Probability & Measure, Wiley, New York
- DasGupta, A. (2008), Asymptotic Theory of Statistics & Probability, Springer, New York
- Ferguson, T. S. (1996), A Course in Large Sample Theory, Chapman & Hall, London
- Le Cam, L. M. and Yang, G. (1990), Asymptotics in Statistics: Some Basic Concepts, Springer,New York
- Lehmann, E. L. (1999), Elements of Large Sample Theory, Springer, New York
- Rajarshi, M. B. (2012), Statistical Inference for Discrete Time Stochastic Processes, Springer,New Delhi
- Roussas, G. G. (1972), Contiguity of Probability Measures: Some Applications in Statistics,Cambridge University Press, London
- van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press, London

**A tentative time-table mentioning the names of speakers and course associates**

Lecture 1 |
Tea |
Lecture 2 |
Lunch |
Tutorial 1 or Guest Lecture |
Tea |
Tutorial 2 or Guest Lecture |
Snacks |
||

Day | Date | 9.30-11.00 | 11.00-11.30 | 11.30 -1.00 | 1.00-2.30 | 2.30 – 3.30 | 3.30-4.00 | 4.00-5.00 | 5.00-5.30 |

1 | 23/10/2017 | VS | VS | MR, SP | MR,SP | ||||

2 | 24/10/2017 | VS | VS | MR, SP | MR, SP | ||||

3 | 25/10/2017 | VS | VS | GL-1 | GL-2 | ||||

4 | 26/10/2017 | BVR | BVR | AM, SP | AM, SP | ||||

5 | 27/10/2017 | BVR | BVR | AM, MR | AM, MR | ||||

6 | 28/10/2017 | BVR | BVR | GL-3 | GL-4 | ||||

7 | 30/10/2017 | SRD | SRD | AM, ASK | AM, ASK | ||||

8 | 31/10/2017 | SRD | UVN | AM, ASK | MGK, ASK | ||||

9 | 01/11/2017 | UVN | UVN | MGK, ASK | MGK, ASK | ||||

10 | 02/11/2017 | TVR | TVR | SP, MR | SP, MR | ||||

11 | 03/11/2017 | TVR | MBR | SP, MR | ASK, MGK | ||||

12 | 04/11/2017 | MBR | MBR | ASK, MGK | ASK, MGK |

MR: Mukund Ramtirthkar, SP: Sagar Pandhare, AM: Anuj Mishra, MGK: Madhuri Kulkarni, ASK: Akanksha Kashikar

GL: Guest Lecture on related topics