Theory of Random Sets (eBook)

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

Preface 7
Contents 13
1 Random Closed Sets and Capacity Functionals 17
1 The Choquet theorem 17
1.1 Set-valued random elements 17
1.2 Capacity functionals 20
1.3 Proofs of the Choquet theorem 29
1.4 Separating classes 34
1.5 Random compact sets 36
1.6 Further functionals related to random sets 38
2 Measurability and selections 41
2.1 Multifunctions in metric spaces 41
2.2 Selections of random closed sets 47
2.3 Measurability of set-theoretic operations 53
2.4 Random closed sets in Polish spaces 56
2.5 Non-closed random sets 57
3 Lattice-theoretic framework 58
3.1 Basic constructions 58
3.2 Existence of measures on partially ordered sets 59
3.3 Locally .nite measures on posets 62
3.4 Existence of random sets distributions 63
4 Capacity functionals and properties of random closed sets 65
4.1 Invariance and stationarity 65
4.2 Separable random sets and inclusion functionals 67
4.3 Regenerative events 72
4.4 Robbins’ theorem 75
4.5 Hausdorff dimension 76
4.6 Random open sets 79
4.7 C-additive capacities and random convex sets 80
4.8 Comparison of random sets 83
5 Calculus with capacities 86
5.1 Choquet integral 86
5.2 The Radon–Nikodym theorem for capacities 88
5.3 Dominating probability measures 91
5.4 Carath´eodory’s extension 93
5.5 Derivatives of capacities 95
6 Convergence 100
6.1 Weak convergence 100
6.2 Convergence almost surely and in probability 106
6.3 Probability metrics 109
7 Random sets and hitting processes 113
7.1 Hitting processes 113
7.2 Trapping systems 115
7.3 Distributions of random convex sets 118
8 Point processes and random measures 121
8.1 Random sets and point processes 121
8.2 A representation of random sets as point processes 128
8.3 Random sets and random measures 131
8.4 Random capacities 133
8.5 Robbin’s theorem for random capacities 135
9 Various interpretations of capacities 140
9.1 Non-additive measures 140
9.2 Belief functions 143
9.3 Upper and lower probabilities 145
9.4 Capacities in robust statistics 148
Notes to Chapter 1 150
2 Expectations of Random Sets 161
1 The selection expectation 161
1.1 Integrable selections 161
1.2 The selection expectation 166
1.3 Applications to characterisation of distributions 176
1.4 Variants of the selection expectation 177
1.5 Convergence of the selection expectations 181
1.6 Conditional expectation 186
2 Further de.nitions of expectations 190
2.1 Linearisation approach 190
2.2 The Vorob’ev expectation 192
2.3 Distance average 194
2.4 Radius-vector expectation 198
3 Expectations on lattices and in metric spaces 199
3.1 Evaluations and expectations on lattices 199
3.2 Fr´echet expectation 200
3.3 Expectations of Doss and Herer 202
3.4 Properties of expectations 206
Notes to Chapter 2 207
3 Minkowski Addition 211
1 Strong law of large numbers for random sets 211
1.1 Minkowski sums of deterministic sets 211
1.2 Strong law of large numbers 214
1.3 Applications of the strong law of large numbers 216
1.4 Non-identically distributed summands 222
1.5 Non-compact summands 225
2 The central limit theorem 229
2.1 A central limit theorem for Minkowski averages 229
2.2 Gaussian random sets 234
2.3 Stable random compact sets 236
2.4 Minkowski in.nitely divisible random compact sets 237
3 Further results related to Minkowski sums 239
3.1 Law of iterated logarithm 239
3.2 Three series theorem 240
3.3 Koml´os theorem 242
3.4 Renewal theorems for random convex compact sets 242
3.5 Ergodic theorems 246
3.6 Large deviation estimates 248
3.7 Convergence of functionals 249
3.8 Convergence of random broken lines 250
3.9 An application to allocation problem 251
3.10 In.nite divisibility in positive convex cones 252
Notes to Chapter 3 253
4 Unions of Random Sets 257
1 Union-in.nite-divisibility and union-stability 257
1.1 Extreme values: a reminder 257
1.2 In.nite divisibility for unions 258
1.3 Union-stable random sets 263
1.4 Other normalisations 269
1.5 In.nite divisibility of lattice-valued random elements 274
2 Weak convergence of normalised unions 278
2.1 Sufficient conditions 278
2.2 Necessary conditions 281
2.3 Scheme of series for unions of random closed sets 285
3 Convergence with probability 1 286
3.1 Regularly varying capacities 286
3.2 Almost sure convergence of scaled unions 288
3.3 Stability and relative stability of unions 291
3.4 Functionals of unions 293
4 Convex hulls 294
4.1 Infinite divisibility with respect to convex hulls 294
4.2 Convex-stable sets 297
4.3 Convergence of normalised convex hulls 300
5 Unions and convex hulls of random functions 302
5.1 Random points 302
5.2 Multivalued mappings 304
6 Probability metrics method 309
6.1 Inequalities between metrics 309
6.2 Ideal metrics and their applications 311
Notes to Chapter 4 315
5 Random Sets and Random Functions 319
1 Random multivalued functions 319
1.1 Multivalued martingales 319
1.2 Set-valued random processes 328
1.3 Random functions with stochastic domains 335
2 Levels and excursion sets of random functions 338
2.1 Excursions of random .elds 338
2.2 Random subsets of the positive half-line and filtrations 341
2.3 Level sets of strong Markov processes 345
2.4 Set-valued stopping times and set-indexed martingales 350
3 Semicontinuous random functions 352
3.1 Epigraphs of random functions and epiconvergence 352
3.2 Stochastic optimisation 364
3.3 Epigraphs and extremal processes 369
3.4 Increasing set-valued processes of excursion sets 377
3.5 Strong law of large numbers for epigraphical sums 379
3.6 Level sums of random upper semicontinuous functions 382
3.7 Graphical convergence of random functions 385
Notes to Chapter 5 394
Appendices 403
A Topological spaces and linear spaces 403
B Space of closed sets 414
C Compact sets and the Hausdorff metric 418
D Multifunctions and semicontinuity 425
E Measures, probabilities and capacities 428
F Convex sets 437
H Regular variation 444
References 451
List of Notation 479
Name Index 483
Subject Index 491

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.