Molchanov Theory of Random Sets

1. Auflage 2005
ISBN: 978-1-84628-150-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 488 Seiten

Reihe: Probability and Its Applications

This is the first systematic exposition of random sets theory since Matheron (1975), with full proofs, exhaustive bibliographies and literature notes Interdisciplinary connections and applications of random sets are emphasized throughout the bookAn extensive bibliography in the book is available on the Web at http://liinwww.ira.uka.de/bibliography/math/random.closed.sets.html, and is accompanied by a search engine

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

Autoren/Hrsg.

Molchanov, Ilya

Inhaltsverzeichnis

1;Preface;7
2;Contents;13
3;1 Random Closed Sets and Capacity Functionals;17
3.1;1 The Choquet theorem;17
3.1.1;1.1 Set-valued random elements;17
3.1.2;1.2 Capacity functionals;20
3.1.3;1.3 Proofs of the Choquet theorem;29
3.1.4;1.4 Separating classes;34
3.1.5;1.5 Random compact sets;36
3.1.6;1.6 Further functionals related to random sets;38
3.2;2 Measurability and selections;41
3.2.1;2.1 Multifunctions in metric spaces;41
3.2.2;2.2 Selections of random closed sets;47
3.2.3;2.3 Measurability of set-theoretic operations;53
3.2.4;2.4 Random closed sets in Polish spaces;56
3.2.5;2.5 Non-closed random sets;57
3.3;3 Lattice-theoretic framework;58
3.3.1;3.1 Basic constructions;58
3.3.2;3.2 Existence of measures on partially ordered sets;59
3.3.3;3.3 Locally .nite measures on posets;62
3.3.4;3.4 Existence of random sets distributions;63
3.4;4 Capacity functionals and properties of random closed sets;65
3.4.1;4.1 Invariance and stationarity;65
3.4.2;4.2 Separable random sets and inclusion functionals;67
3.4.3;4.3 Regenerative events;72
3.4.4;4.4 Robbins’ theorem;75
3.4.5;4.5 Hausdorff dimension;76
3.4.6;4.6 Random open sets;79
3.4.7;4.7 C-additive capacities and random convex sets;80
3.4.8;4.8 Comparison of random sets;83
3.5;5 Calculus with capacities;86
3.5.1;5.1 Choquet integral;86
3.5.2;5.2 The Radon–Nikodym theorem for capacities;88
3.5.3;5.3 Dominating probability measures;91
3.5.4;5.4 Carath´eodory’s extension;93
3.5.5;5.5 Derivatives of capacities;95
3.6;6 Convergence;100
3.6.1;6.1 Weak convergence;100
3.6.2;6.2 Convergence almost surely and in probability;106
3.6.3;6.3 Probability metrics;109
3.7;7 Random sets and hitting processes;113
3.7.1;7.1 Hitting processes;113
3.7.2;7.2 Trapping systems;115
3.7.3;7.3 Distributions of random convex sets;118
3.8;8 Point processes and random measures;121
3.8.1;8.1 Random sets and point processes;121
3.8.2;8.2 A representation of random sets as point processes;128
3.8.3;8.3 Random sets and random measures;131
3.8.4;8.4 Random capacities;133
3.8.5;8.5 Robbin’s theorem for random capacities;135
3.9;9 Various interpretations of capacities;140
3.9.1;9.1 Non-additive measures;140
3.9.2;9.2 Belief functions;143
3.9.3;9.3 Upper and lower probabilities;145
3.9.4;9.4 Capacities in robust statistics;148
3.10;Notes to Chapter 1;150
4;2 Expectations of Random Sets;161
4.1;1 The selection expectation;161
4.1.1;1.1 Integrable selections;161
4.1.2;1.2 The selection expectation;166
4.1.3;1.3 Applications to characterisation of distributions;176
4.1.4;1.4 Variants of the selection expectation;177
4.1.5;1.5 Convergence of the selection expectations;181
4.1.6;1.6 Conditional expectation;186
4.2;2 Further de.nitions of expectations;190
4.2.1;2.1 Linearisation approach;190
4.2.2;2.2 The Vorob’ev expectation;192
4.2.3;2.3 Distance average;194
4.2.4;2.4 Radius-vector expectation;198
4.3;3 Expectations on lattices and in metric spaces;199
4.3.1;3.1 Evaluations and expectations on lattices;199
4.3.2;3.2 Fr´echet expectation;200
4.3.3;3.3 Expectations of Doss and Herer;202
4.3.4;3.4 Properties of expectations;206
4.4;Notes to Chapter 2;207
5;3 Minkowski Addition;211
5.1;1 Strong law of large numbers for random sets;211
5.1.1;1.1 Minkowski sums of deterministic sets;211
5.1.2;1.2 Strong law of large numbers;214
5.1.3;1.3 Applications of the strong law of large numbers;216
5.1.4;1.4 Non-identically distributed summands;222
5.1.5;1.5 Non-compact summands;225
5.2;2 The central limit theorem;229
5.2.1;2.1 A central limit theorem for Minkowski averages;229
5.2.2;2.2 Gaussian random sets;234
5.2.3;2.3 Stable random compact sets;236
5.2.4;2.4 Minkowski in.nitely divisible random compact sets;237
5.3;3 Further results related to Minkowski sums ;239
5.3.1;3.1 Law of iterated logarithm;239
5.3.2;3.2 Three series theorem;240
5.3.3;3.3 Koml´os theorem;242
5.3.4;3.4 Renewal theorems for random convex compact sets;242
5.3.5;3.5 Ergodic theorems;246
5.3.6;3.6 Large deviation estimates;248
5.3.7;3.7 Convergence of functionals;249
5.3.8;3.8 Convergence of random broken lines;250
5.3.9;3.9 An application to allocation problem;251
5.3.10;3.10 In.nite divisibility in positive convex cones;252
5.4;Notes to Chapter 3;253
6;4 Unions of Random Sets;257
6.1;1 Union-in.nite-divisibility and union-stability;257
6.1.1;1.1 Extreme values: a reminder;257
6.1.2;1.2 In.nite divisibility for unions;258
6.1.3;1.3 Union-stable random sets;263
6.1.4;1.4 Other normalisations;269
6.1.5;1.5 In.nite divisibility of lattice-valued random elements;274
6.2;2 Weak convergence of normalised unions;278
6.2.1;2.1 Sufficient conditions;278
6.2.2;2.2 Necessary conditions;281
6.2.3;2.3 Scheme of series for unions of random closed sets;285
6.3;3 Convergence with probability 1;286
6.3.1;3.1 Regularly varying capacities;286
6.3.2;3.2 Almost sure convergence of scaled unions;288
6.3.3;3.3 Stability and relative stability of unions;291
6.3.4;3.4 Functionals of unions;293
6.4;4 Convex hulls;294
6.4.1;4.1 Infinite divisibility with respect to convex hulls;294
6.4.2;4.2 Convex-stable sets;297
6.4.3;4.3 Convergence of normalised convex hulls;300
6.5;5 Unions and convex hulls of random functions;302
6.5.1;5.1 Random points;302
6.5.2;5.2 Multivalued mappings;304
6.6;6 Probability metrics method;309
6.6.1;6.1 Inequalities between metrics;309
6.6.2;6.2 Ideal metrics and their applications;311
6.7;Notes to Chapter 4;315
7;5 Random Sets and Random Functions;319
7.1;1 Random multivalued functions;319
7.1.1;1.1 Multivalued martingales;319
7.1.2;1.2 Set-valued random processes;328
7.1.3;1.3 Random functions with stochastic domains;335
7.2;2 Levels and excursion sets of random functions;338
7.2.1;2.1 Excursions of random .elds;338
7.2.2;2.2 Random subsets of the positive half-line and filtrations;341
7.2.3;2.3 Level sets of strong Markov processes;345
7.2.4;2.4 Set-valued stopping times and set-indexed martingales;350
7.3;3 Semicontinuous random functions;352
7.3.1;3.1 Epigraphs of random functions and epiconvergence;352
7.3.2;3.2 Stochastic optimisation;364
7.3.3;3.3 Epigraphs and extremal processes;369
7.3.4;3.4 Increasing set-valued processes of excursion sets;377
7.3.5;3.5 Strong law of large numbers for epigraphical sums;379
7.3.6;3.6 Level sums of random upper semicontinuous functions;382
7.3.7;3.7 Graphical convergence of random functions;385
7.4;Notes to Chapter 5;394
8;Appendices;403
8.1;A Topological spaces and linear spaces;403
8.2;B Space of closed sets;414
8.3;C Compact sets and the Hausdorff metric;418
8.4;D Multifunctions and semicontinuity;425
8.5;E Measures, probabilities and capacities;428
8.6;F Convex sets;437
8.7;H Regular variation;444
9;References;451
10;List of Notation;479
11;Name Index;483
12;Subject Index;491

Leseproben

2 Expectations of Random Sets (p.145)

1 The selection expectation

The space F of closed sets (and also the space K of compact sets) is non-linear, so that conventional concepts of expectations in linear spaces are not directly applicable for random closed (or compact) sets. Sets have different features (that often are dif.cult to express numerically) and particular definitions of expectations highlight various features important in the chosen context.

To explain that an expectation of a random closed (or compact) set is not straightforward to de.ne, consider a random closed set X which equals [0, 1] with probability 1/2 and otherwise is {0, 1}. For another example, let X be a triangle with probability 1/2 and a disk otherwise. A "reasonable" expectation in either example is not easy to de.ne. Strictly speaking, the de.nition of the expectation depends on what the objective is, which features of random sets are important to average and which are possible to neglect.

This section deals with the selection expectation (also called the Aumann expectation), which is the best investigated concept of expectation for random sets. Since many results can be naturally formulated for random closed sets in Banach spaces, we assume that E is a separable Banach space unless stated otherwise. Special features inherent to expectations of random closed sets in Rd will be highlighted throughout. To avoid unnecessary complications, it is always assumed that all random closed sets are almost surely non-empty.

1.1 Integrable selections

The key idea in the de.nition of the selection expectation is to represent a random closed set as a family of its integrable selections. The concept of a selection of a random closed set was introduced in De.nition 1.2.2.While properties of selections discussed in Section 1.2.1 can be formulated without assuming a linear structure on E, now we discuss further features of random selections with the key issue being their integrability.