Chains of Infinite Order [MA6011]
Wintersemester 2021/20
Prof. Dr. Noam Berger and Dr. Diana Conache
- Location: To be specified later
- Prerequisites: Einführung in die Wahrscheinlichkeitstheorie und Statistik [MA0009], Maß- und Integrationstheorie [MA2003] and highly recommended also Markov Chains [MA2404].
- 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
- 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
- A. Galves, E. Löcherbach - Stochastic chains with memory of variable length
- N. Berger, C. Hoffman, V. Sidoravicius - Nonuniqueness for specifications in ℓ^{2+ε}
- Manfred Lehn - Wie halte ich einen Seminarvortrag
- Talks:
- k-step Markov Chains
- g-measures: Definitions, regularity, existence and examples
- Convex structure of the set of g-measures
- Square summability condition for uniqueness
- Square summability condition for uniqueness is tight
- Non-uniqueness: The Bramson-Kalikow Model
- Markov approximation
- Regenerative construction and perfect simulation
- Variable length Markov Chains
- Bachelor or Master Thesis: You will also have the possibility to write a Bachelor or a Master thesis after this seminar. For more details please contact us.