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E-raamat: Theory of Random Sets

Ilya Molchanov

Formaat: PDF+DRM
Sari: Probability Theory and Stochastic Modelling 87
Ilmumisaeg: 14-Dec-2017
Kirjastus: Springer London Ltd
Keel: eng
ISBN-13: 9781447173496

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Formaat: PDF+DRM
Sari: Probability Theory and Stochastic Modelling 87
Ilmumisaeg: 14-Dec-2017
Kirjastus: Springer London Ltd
Keel: eng
ISBN-13: 9781447173496

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This monograph, now in a thoroughly revised second edition, offers the latest research on random sets. It has been extended to include substantial developments achieved since 2005, some of them motivated by applications of random sets to econometrics and finance.

The present volume builds on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time fixes terminology and notation that often vary in the literature, establishing it as a natural part of modern probability theory and providing a platform for future development. It is completely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight.

Aimed at research level, Theory of Random Sets will be an invaluable reference for probabilists; mathematicians working in convex and integral geometry, set-valued analysis, capacity and potential theory; mathematical statisticians in spatial statistics and uncertainty quantification; specialists in mathematical economics, econometrics, decision theory, and mathematical finance; and electronic and electrical engineers interested in image analysis.

Arvustused

A strong feature of the first edition that is further strengthened in the second one is the Notes concluding each chapter. These remarks and discussions not only provide a comprehensive literature review on the historical origins of the material covered in each section, but also point to a variety of recent research papers, where those concepts are applied. (Christian Hirsch, Mathematical Reviews, August, 2018)

1 Random Closed Sets and Capacity Functionals

(224)

1.1 Distributions of Random Sets

(48)

1.1.1 Set-Valued Random Elements

(5)

1.1.2 Capacity Functionals

(9)

1.1.3 Choquet's Theorem

(5)

1.1.4 Proofs of Choquet's Theorem

(7)

1.1.5 Separating Classes

(4)

1.1.6 Further Functionals Related to Random Sets

(7)

1.1.7 Separable Random Sets and Inclusion Functionals

(6)

1.1.8 Hitting Processes

(5)

1.2 The Lattice-Theoretic Framework

(8)

1.2.1 Basic Constructions

(1)

1.2.2 Existence of Measures on Partially Ordered Sets

(4)

1.2.3 Locally Finite Measures on Posets

(2)

1.2.4 Existence of Random Sets Distributions

(1)

1.3 Measurability and Multifunctions

(20)

1.3.1 Multifunctions in Metric Spaces

(5)

1.3.2 Random Compact Sets in Polish Spaces

(2)

1.3.3 The Effros σ-Algebra

(3)

1.3.4 Distribution of Random Closed Sets in Polish Spaces

(2)

1.3.5 Measurability of Set-Theoretic Operations

(3)

1.3.6 Non-closed Random Sets

(5)

1.4 Selections of Random Closed Sets

(13)

1.4.1 Existence and Uniqueness

(5)

1.4.2 Distributions of Selections

(6)

1.4.3 Families of Selections

(2)

1.5 Capacity Functionals and Properties of Random Closed Sets

(22)

1.5.1 Invariance and Stationarity

(4)

1.5.2 Regenerative Events

(3)

1.5.3 The Expected Measure of a Random Set

(3)

1.5.4 Hausdorff Dimension

100

(4)

1.5.5 Comparison of Random Sets

104

(4)

1.5.6 Transformation of Capacities

108

(2)

1.5.7 Rearrangement Invariance

110

(2)

1.6 Calculus with Capacities

112

(15)

1.6.1 The Choquet Integral

112

(8)

1.6.2 The Radon--Nikodym Theorem for Capacities

120

(2)

1.6.3 Derivatives of Capacities

122

(5)

1.7 Convergence

127

(19)

1.7.1 Weak Convergence

127

(9)

1.7.2 Convergence Almost Surely and in Probability

136

(4)

1.7.3 Probability Metrics

140

(6)

1.8 Random Convex Sets

146

(15)

1.8.1 C-Additive Capacities

146

(4)

1.8.2 Containment Functional

150

(6)

1.8.3 Non-compact Random Convex Sets

156

(5)

1.9 Point Processes and Random Measures

161

(27)

1.9.1 Random Sets and Point Processes

161

(9)

1.9.2 A Representation of Random Sets as Point Processes

170

(5)

1.9.3 Random Sets and Random Measures

175

(3)

1.9.4 Random Capacities

178

(3)

1.9.5 Robbins' Theorem for Random Capacities

181

(7)

1.10 Various Interpretations of Capacities

188

(37)

1.10.1 Non-additive Measures

188

(4)

1.10.2 Upper and Lower Probabilities

192

(6)

1.10.3 Belief Functions

198

(3)

1.10.4 Capacities in Robust Statistics

201

(3)

Notes to Chap. 1

204

(21)

2 Expectations of Random Sets

225

(92)

2.1 The Selection Expectation and Aumann Integral

225

(53)

2.1.1 Integrable Selections and Decomposability

226

(12)

2.1.2 The Selection Expectation

238

(13)

2.1.3 Applications of the Selection Expectation

251

(8)

2.1.4 Variants of the Selection Expectation

259

(4)

2.1.5 Convergence of the Selection Expectations

263

(7)

2.1.6 Conditional Expectation

270

(8)

2.2 Further Definitions of Expectations

278

(39)

2.2.1 General Methods of Defining Expectations

278

(4)

2.2.2 The Vorob'ev Expectation

282

(4)

2.2.3 Distance Average

286

(4)

2.2.4 The Radius-Vector Expectation

290

(1)

2.2.5 Expectations in Metric Spaces

291

(7)

2.2.6 Convex Combination Spaces

298

(1)

2.2.7 Sublinear and Superlinear Expectations

299

(7)

Notes to Chap. 2

306

(11)

3 Minkowski Sums

317

(62)

3.1 The Strong Law of Large Numbers for Random Sets

317

(27)

3.1.1 Minkowski Sums of Deterministic Sets

317

(3)

3.1.2 The Strong Law of Large Numbers for Random Compact Sets

320

(6)

3.1.3 Applications of the Strong Law of Large Numbers

326

(9)

3.1.4 Non-identically Distributed Summands

335

(4)

3.1.5 Non-compact Summands

339

(5)

3.2 Limit Theorems

344

(17)

3.2.1 The Central Limit Theorem for Minkowski Averages

344

(6)

3.2.2 Gaussian Random Sets

350

(4)

3.2.3 Minkowski Infinitely Divisible Random Compact Sets

354

(3)

3.2.4 Stable Random Compact Sets

357

(4)

3.3 Further Results Related to Minkowski Sums

361

(18)

3.3.1 Convergence of Series

361

(2)

3.3.2 Renewal Theorems

363

(4)

3.3.3 Ergodic Theorems

367

(2)

3.3.4 Large Deviations

369

(4)

Notes to Chap. 3

373

(6)

4 Unions of Random Sets

379

(72)

4.1 Infinite Divisibility and Stability for Unions

379

(30)

4.1.1 Infinite Divisibility for Unions

379

(7)

4.1.2 Scheme of Series for Unions of Random Closed Sets

386

(2)

4.1.3 Infinite Divisibility of Lattice-Valued Random Elements

388

(4)

4.1.4 Union-Stable Random Sets

392

(6)

4.1.5 LePage Series Representation and Examples

398

(6)

4.1.6 Non-multiplicative Normalisations

404

(5)

4.2 Weak Convergence of Scaled Unions

409

(15)

4.2.1 Stability of Limits

409

(1)

4.2.2 Limit Theorems Under Regular Variation Conditions

410

(7)

4.2.3 Necessary Conditions

417

(3)

4.2.4 The Probability Metrics Method

420

(4)

4.3 Convergence with Probability One

424

(10)

4.3.1 Regularly Varying Capacities

424

(2)

4.3.2 Almost Sure Convergence of Scaled Unions

426

(3)

4.3.3 Unions of Random Compact Sets

429

(3)

4.3.4 Functionals of Unions

432

(2)

4.4 Convex Hulls and Intersections

434

(17)

4.4.1 Infinite Divisibility for Convex Hulls

434

(3)

4.4.2 Convex-Stable Sets

437

(3)

4.4.3 Intersections

440

(4)

Notes to Chap. 4

444

(7)

5 Random Sets and Random Functions

451

(102)

5.1 Random Multivalued Functions

451

(35)

5.1.1 Multivalued Martingales in Discrete Time

451

(11)

5.1.2 Continuous Time Set-Valued Processes

462

(11)

5.1.3 Special Classes of Set-Valued Processes

473

(9)

5.1.4 Random Functions with Stochastic Domains

482

(4)

5.2 Level and Excursion Sets of Random Functions

486

(17)

5.2.1 Excursions of Random Fields

486

(5)

5.2.2 Random Subsets of the Positive Half-Line and Filtrations

491

(3)

5.2.3 Level Sets of Strong Markov Processes

494

(7)

5.2.4 Set-Valued Stopping Times and Set-Indexed Martingales

501

(2)

5.3 Semicontinuous Random Functions

503

(50)

5.3.1 Epigraphs and Epi-Convergence

503

(4)

5.3.2 Weak Epi-Convergence of Random Functions

507

(11)

5.3.3 Stochastic Optimisation

518

(5)

5.3.4 Epigraphs and Extremal Processes

523

(9)

5.3.5 Increasing Set-Valued Processes of Excursion Sets

532

(2)

5.3.6 Strong Law of Large Numbers for Epigraphical Sums

534

(3)

5.3.7 Level Sums of Random Upper Semicontinuous Functions

537

(3)

Notes to Chap. 5

540

(13)

Appendices

553

(60)

A Topological Spaces and Metric Spaces

553

(7)

B Linear Spaces

560

(6)

C Space of Closed Sets

566

(5)

D Compact Sets and the Hausdorff Metric

571

(8)

E Multifunctions and Semicontinuity

579

(4)

F Measures and Probabilities

583

(7)

G Capacities

590

(5)

H Convex Sets

595

(7)

I Semigroups, Cones and Harmonic Analysis

602

(4)

J Regular Variation

606

(7)

References

613

(36)

Name Index

649

(10)

Subject Index

659

(16)

List of Notation

675

Ilya Molchanov is Professor of Probability Theory at the Department of Mathematical Statistics and Actuarial Science at the University of Bern, Switzerland.

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