Cover Page
The handle http://hdl.handle.net/1887/137985 holds various files of this Leiden University
dissertation.
Author:
Berghout, S.
Gibbs Processes and Applications
Proefschrift
ter verkrijging
van de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker,
volgens besluit van het College voor Promoties te verdedigen op dinsdag 27 oktober 2020
klokke 13.45 uur
door
Steven Berghout
Promotoren:
Prof. dr. E.A. Verbitskiy (Universiteit Leiden & Rijksniversiteit Groningen) Prof. dr. W. Th. F. den Hollander
Samenstelling van de promotiecommissie Prof. dr. P.D. Grunwald
Prof. dr. R.M. van Luijk
Prof. dr. A.C.D. van Enter (Rijksuniversiteit Groningen) Prof. dr. C. Külske (Ruhr-Universität Bochum)
Prof. dr. A. Le Ny (Université Gustave-Eiffel, Paris)
Contents
1 Introduction 5
1.1 Finite-range dependence models: Markov processes and fields . . . 6
1.2 Long-range dependence models . . . 7
1.3 Overview of the main results . . . 11
2 On the relation between Gibbs and g-measures 19 2.1 Introduction . . . 19
2.2 Four classes of measures . . . 20
2.3 Further properties of Gibbs measures . . . 26
2.4 Main results . . . 29
2.5 Examples and an Overview . . . 39
2.6 When Gibbs measures are g-measures . . . 47
2.7 Concluding remarks . . . 50
3 Estimation of two-sided conditional probabilities 51 3.1 Unidirectional modeling . . . 52
3.2 Bidirectional modeling . . . 56
3.3 Gibbs measures . . . 58
3.4 Conclusions . . . 80
4 On regularity of functions of Markov chains 83 4.1 Introduction . . . 83
4
4.2 Properties of factors of Markov measures . . . 87
4.3 Continuous measure disintegrations . . . 91
4.4 Thermodynamic formalism for fibred systems . . . 97
4.5 Examples . . . 108
5 Renormalisation of Gibbs states 113 5.1 Introduction . . . 113
5.2 Fuzzy Gibbs states . . . 116
5.3 Conditional measures on fibres . . . 122
5.4 Tjur points . . . 126
5.5 Limiting conditional distributions . . . 132
5.6 Conclusions and Outlook . . . 139
Bibliography 141
Samenvatting 151
Acknowledgements 153
