Random Measures Theory and Applications 1st Edition by Olav Kallenberg 3319415987 9783319415987

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ISBN-10 :  3319415987 

ISBN-13 : 9783319415987

Author:  Olav Kallenberg 

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.

Random Measures Theory and Applications 1st Table of contents:

1 Spaces, Kernels, and Disintegration
1.1 Borel and Measure Spaces
1.2 Product and Factorial Measures
1.3 Kernels and Operators
1.4 Disintegration
1.5 Differentiation
2 Distributions and Local Structure
2.1 Uniqueness, Intensities, and Regularity
2.2 Absolute Continuity and Conditioning
2.3 Additive and Maxitive Processes
3 Poisson and Related Processes
3.1 Basic Processes and Uniqueness Criteria
3.2 Linear Poisson and Binomial Processes
3.3 Independence and Infinite Divisibility
3.4 Poisson and Related Integrals
3.5 Symmetric Sequences and Processes
4 Convergence and Approximation
4.1 Weak and Vague Topologies
4.2 Convergence in Distribution
4.3 Null Arrays and Infinite Divisibility
4.4 Strong Approximation and Convergence
4.5 Transforms and Symmetries
5 Stationarity in Euclidean Spaces
5.1 Palm Measures and Cycle Stationarity
5.2 Inversion and Spacing Measures
5.3 Asymptotic Invariance
5.4 Averaging and Smoothing Limits
5.5 Palm and Spacing Averages
5.6 Local Invariance
5.7 Ballot Theorems and Sojourn Laws
6 Palm and Related Kernels
6.1 Campbell Measures and Palm Kernels
6.2 Reduced Palm Measures and Conditioning
6.3 Slivnyak Properties and Factorization
6.4 Iterated Conditioning and Palm Recursion
6.5 Randomizations and Cox Processes
6.6 Local Hitting and Conditioning
6.7 Duality and Kernel Representation
7 Group Stationarity and Invariance
7.1 Invariant Measures and Kernels
7.2 Invariant Representations and Palm Kernels
7.3 Measure Inversion
7.4 Duality and Mass Transport
7.5 Invariant Disintegration
7.6 Stationary Densities and Disintegration
8 Exterior Conditioning
8.1 Gibbs and Papangelou Kernels
8.2 Transformation Properties
8.3 Regularity Conditions
8.4 Recursion and Symmetry
8.5 Local Conditioning and Decomposition
8.6 External Intensity and Projection
9 Compensation and Time Change
9.1 Predictable Times and Processes
9.2 Doob–Meyer Decomposition and Compensation
9.3 Predictable Invariance and Time Change
9.4 Discounted Compensator and Predictable Maps
9.5 Extended Compensator and Integral Representation
9.6 Tangential Existence
9.7 Tangential Comparison
10 Multiple Integration
10.1 Poisson and Related Integrals
10.2 Symmetric Point-process Integrals
10.3 Escape Criteria
10.4 L´evy and Related Integrals
10.5 Multiple Series and Integrals
10.6 Tangential Reduction and Decoupling
11 Line and Flat Processes
11.1 Stationary Line Processes in the Plane
11.2 A non-Cox Counterexample
11.3 Spanning Criteria for Invariance
11.4 Invariance under Absolute Continuity
11.5 Non-Interactive Particle Systems
11.6 Degeneracies of Flat Processes
11.7 General Criteria of Absolute Continuity
12 Regeneration and Local Time
12.1 Renewal and Occupation Measures
12.2 Excursion Local Time and Point Process
12.3 Semi-Martingale Local Time
12.4 Moment and Palm Measures
12.5 Density Existence and Continuity
12.6 Conditioning via Duality
12.7 Regular Palm Distributions
12.8 Local Hitting and Conditioning
13 Branching Systems and Super-processes
13.1 Binary Splitting and Scaling Limits
13.2 Finiteness, Extinction, and Genealogy
13.3 Moment Measures and Palm Trees
13.4 Moment Densities
13.5 Regular Palm Distributions
13.6 Hitting Rates and Approximation
13.7 Lebesgue Approximation
13.8 Local Stationarity and Invariant Cluster
13.9 Local Hitting and Conditioning
13.10 Stability of Discrete Clustering

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Tags: Random Measures, Theory, Applications, Olav Kallenberg