Chains of Infinite Order [MA6011]
Wintersemester 2017/18
Dr. Stein Andreas Bethuelsen and Dr. Diana Conache
- Location: Glaskasten 03.12.020A
- Prerequisites: Probability theory [MA2409] and highly recommended also Markov Chains [MA2404]. Note that students registered for Markovketten [MA2404] during the winter semester 2017/2018 are welcome to join.
- Contents: Chains of infinite order are the generalization of Markov chains, where the process may depend on its entire history. For such chains, which are often more natural for many applications, new and interesting phenomena arise. This seminar will firstly introduce students to the foundation of this theory by considering questions such as existence and uniqueness of invariant measures, and also the possibility of phase transitions. In the second part of the seminar the focus will be on a probabilistic description of the theory. The students will be introduced to certain coupling techniques which in turn will be used to answer the following two questions:
- When can a chain of infinite order be approximated by a Markov chain?
- When is it possible to sample the stationary distribution exactly (by a perfect sampling algorithm)?
- Bibliography:
- S. Friedli - Lecture Notes: On the specifications of probabilities by regular g-functions
- R. Fernández and A. Galves - Markov approximations of chains of infinite order
- F. Comets, R. Fernandez, P. A. Ferrari- Processes with long memory: Regenerative construction and perfect simulation
- S. Friedli - A note on the Bramson-Kalikov Process
- E. Verbitskiy - Lecture Notes: Hidden Gibbs models. Theory and applications
- Manfred Lehn - Wie halte ich einen Seminarvortrag Optional:
- M. Bramsow, S. Kalikow -Non-uniqueness in g-functions, Israel J. Math. 84 (1993), no. 1-2, 153–160
- H. Berbee - Chains with inifinite connections: uniqueness and Markov representation, Probability Theory and Related Fields, Vol. 76, No. 2, 1987, pp. 243-253. doi:10.1007/BF00319986
- R. Fernandez, P. A. Ferrari, A. Galves - Coupling, renewal and perfect simulation of chains of infinite order
- G. Maillard - Introduction to Chains with Complete Connections
- Schedule of the talks:
- Day 1 Monday, 4.12.2017
- 12:00-12:45 Introductory lecture - Stein Andreas and Diana
- 12:45-14:00 Definitions, regularity, existence and examples
- 14:15-15:30 Structure of the set of g-measures
- 15:45-17:00 Applications: hidden Markov models
- Day 2 Monday, 18.12.2017
- 12:30-13:45 Markov approximation
- 14:00-15:15 Regenerative construction and perfect simulation
- 15:30-16:45 Square summable condition for uniqueness
- 17:00-18:15 Non-uniqueness
