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2017 | Buch

Spaces, Kernels, and Disintegration Kapitel lesen Erstes Kapitel lesen

Random Measures, Theory and Applications

verfasst von: Olav Kallenberg

Verlag: Springer International Publishing

Buchreihe : Probability Theory and Stochastic Modelling

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Offering the first comprehensive treatment of the theory of random measures, this book has a very broad scope, ranging from basic properties of Poisson and related processes to the modern theories of convergence, stationarity, Palm measures, conditioning, and compensation. The three large final chapters focus on applications within the areas of stochastic geometry, excursion theory, and branching processes. Although this theory plays a fundamental role in most areas of modern probability, much of it, including the most basic material, has previously been available only in scores of journal articles. The book is primarily directed towards researchers and advanced graduate students in stochastic processes and related areas.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Spaces, Kernels, and Disintegration
Abstract
Borel and measure spaces; Product and factorial measures; Kernels and operators; Disintegration; Differentiation.
Olav Kallenberg
Chapter 2. Distributions and Local Structure
Abstract
Uniqueness, intensities, and regularity; Absolute continuity and conditioning; Additive and maxitive processes.
Olav Kallenberg
Chapter 3. Poisson and Related Processes
Abstract
Basic processes and uniqueness criteria; Linear Poisson and binomial processes; Independence and infinite divisibility; Poisson and related integrals; Symmetric sequences and processes.
Olav Kallenberg
Chapter 4. Convergence and Approximation
Abstract
Weak and vague topologies; Convergence in distribution; Null arrays and infinite divisibility; Strong approximation and convergence; Transforms and symmetries.
Olav Kallenberg
Chapter 5. Stationarity in Euclidean Spaces
Abstract
Palm measures and cycle stationarity; Inversion and spacing measures; Asymptotic invariance; Averaging and smoothing limits; Palm and spacing averages; Local invariance; Ballot theorems and sojourn laws.
Olav Kallenberg
Chapter 6. Palm and Related Kernels
Abstract
Campbell measures and Palm kernels; Reduced Palm measures and conditioning; Slivnyak properties and factorization; Iterated conditioning and Palm recursion; Randomizations and Cox processes; Local hitting and conditioning; Duality and kernel representation.
Olav Kallenberg
Chapter 7. Group Stationarity and Invariance
Abstract
Invariant measures and kernels; Invariant representations and Palm kernels; Measure inversion; Duality and mass transport; Invariant disintegration; Stationary densities and disintegration.
Olav Kallenberg
Chapter 8. Exterior Conditioning
Abstract
Gibbs and Papangelou kernels; Transformation properties; Regularity conditions; Recursion and symmetry; Local conditioning and decomposition; External intensity and projection.
Olav Kallenberg
Chapter 9. Compensation and Time Change
Abstract
Predictable times and processes; Doob–Meyer decomposition and compensation; Predictable invariance and time change; Discounted compensator and predictable maps; Extended compensator and integral representation; Tangential existence; Tangential comparison.
Olav Kallenberg
Chapter 10. Multiple Integration
Abstract
Poisson and related integrals; Symmetric point-process integrals; Escape criteria; Lévy and related integrals; Multiple series and integrals; Tangential comparison and decoupling.
Olav Kallenberg
Chapter 11. Line and Flat Processes
Abstract
Stationary line processes in the plane; A non-Cox counterexample; Spanning criteria for invariance; Invariance under absolute continuity; Non-interactive particle systems; Degeneracies of flat processes; General criteria of absolute continuity.
Olav Kallenberg
Chapter 12. Regeneration and Local Time
Abstract
Renewal and occupation measures; Excursion local time and point process; Semi-martingale local time; Moment and Palm measures; Density existence and continuity; Conditioning via duality; Regular Palm distributions; Local hitting and conditioning.
Olav Kallenberg
Chapter 13. Branching Systems and Super-processes
Abstract
Binary splitting and scaling limits; Finiteness, extinction, and genealogy; Moment measures and Palm trees; Moment densities; Regular Palm distributions; Hitting rates and approximation; Lebesgue approximation; Local stationarity and invariant cluster; Local hitting and conditioning; Stability of discrete clustering.
Olav Kallenberg
Backmatter
Metadaten
Titel
Random Measures, Theory and Applications
verfasst von
Olav Kallenberg
Copyright-Jahr
2017
Electronic ISBN
978-3-319-41598-7
Print ISBN
978-3-319-41596-3
DOI
https://doi.org/10.1007/978-3-319-41598-7