Theory of Random Sets (Probability and its Applications)

Contents......Page 13
1.1 Set-valued random elements......Page 17
1.2 Capacity functionals......Page 20
1.3 Proofs of the Choquet theorem......Page 29
1.4 Separating classes......Page 34
1.5 Random compact sets......Page 36
1.6 Further functionals related to random sets......Page 38
2.1 Multifunctions in metric spaces......Page 41
2.2 Selections of random closed sets......Page 47
2.3 Measurability of set-theoretic operations......Page 53
2.4 Random closed sets in Polish spaces......Page 56
2.5 Non-closed random sets......Page 57
3.1 Basic constructions......Page 58
3.2 Existence of measures on partially ordered sets......Page 59
3.3 Locally finite measures on posets......Page 62
3.4 Existence of random sets distributions......Page 63
4.1 Invariance and stationarity......Page 65
4.2 Separable random sets and inclusion functionals......Page 67
4.3 Regenerative events......Page 72
4.4 Robbins’ theorem......Page 75
4.5 Hausdorff dimension......Page 76
4.6 Random open sets......Page 79
4.7 C-additive capacities and random convex sets......Page 80
4.8 Comparison of random sets......Page 83
5.1 Choquet integral......Page 86
5.2 The Radon–Nikodym theorem for capacities......Page 88
5.3 Dominating probability measures......Page 91
5.4 Carathéodory’s extension......Page 93
5.5 Derivatives of capacities......Page 95
6.1 Weak convergence......Page 100
6.2 Convergence almost surely and in probability......Page 106
6.3 Probability metrics......Page 109
7.1 Hitting processes......Page 113
7.2 Trapping systems......Page 115
7.3 Distributions of random convex sets......Page 118
8.1 Random sets and point processes......Page 121
8.2 A representation of random sets as point processes......Page 128
8.3 Random sets and random measures......Page 131
8.4 Random capacities......Page 133
8.5 Robbin’s theorem for random capacities......Page 135
9.1 Non-additive measures......Page 140
9.2 Belief functions......Page 143
9.3 Upper and lower probabilities......Page 145
9.4 Capacities in robust statistics......Page 148
Notes to Chapter 1......Page 150
1.1 Integrable selections......Page 161
1.2 The selection expectation......Page 166
1.3 Applications to characterisation of distributions......Page 176
1.4 Variants of the selection expectation......Page 177
1.5 Convergence of the selection expectations......Page 181
1.6 Conditional expectation......Page 186
2.1 Linearisation approach......Page 190
2.2 The Vorob’ev expectation......Page 192
2.3 Distance average......Page 194
2.4 Radius-vector expectation......Page 198
3.1 Evaluations and expectations on lattices......Page 199
3.2 Fréchet expectation......Page 200
3.3 Expectations of Doss and Herer......Page 202
3.4 Properties of expectations......Page 206
Notes to Chapter 2......Page 207
1.1 Minkowski sums of deterministic sets......Page 211
1.2 Strong law of large numbers......Page 214
1.3 Applications of the strong law of large numbers......Page 216
1.4 Non-identically distributed summands......Page 222
1.5 Non-compact summands......Page 225
2.1 A central limit theorem for Minkowski averages......Page 229
2.2 Gaussian random sets......Page 234
2.3 Stable random compact sets......Page 236
2.4 Minkowski infinitely divisible random compact sets......Page 237
3.1 Law of iterated logarithm......Page 239
3.2 Three series theorem......Page 240
3.4 Renewal theorems for random convex compact sets......Page 242
3.5 Ergodic theorems......Page 246
3.6 Large deviation estimates......Page 248
3.7 Convergence of functionals......Page 249
3.8 Convergence of random broken lines......Page 250
3.9 An application to allocation problem......Page 251
3.10 Infinite divisibility in positive convex cones......Page 252
Notes to Chapter 3......Page 253
1.1 Extreme values: a reminder......Page 257
1.2 Infinite divisibility for unions......Page 258
1.3 Union-stable random sets......Page 263
1.4 Other normalisations......Page 269
1.5 Infinite divisibility of lattice-valued random elements......Page 274
2.1 Sufficient conditions......Page 278
2.2 Necessary conditions......Page 281
2.3 Scheme of series for unions of random closed sets......Page 285
3.1 Regularly varying capacities......Page 286
3.2 Almost sure convergence of scaled unions......Page 288
3.3 Stability and relative stability of unions......Page 291
3.4 Functionals of unions......Page 293
4.1 Infinite divisibility with respect to convex hulls......Page 294
4.2 Convex-stable sets......Page 297
4.3 Convergence of normalised convex hulls......Page 300
5.1 Random points......Page 302
5.2 Multivalued mappings......Page 304
6.1 Inequalities between metrics......Page 309
6.2 Ideal metrics and their applications......Page 311
Notes to Chapter 4......Page 315
1.1 Multivalued martingales......Page 319
1.2 Set-valued random processes......Page 328
1.3 Random functions with stochastic domains......Page 335
2.1 Excursions of random fields......Page 338
2.2 Random subsets of the positive half-line and filtrations......Page 341
2.3 Level sets of strong Markov processes......Page 345
2.4 Set-valued stopping times and set-indexed martingales......Page 350
3.1 Epigraphs of random functions and epiconvergence......Page 352
3.2 Stochastic optimisation......Page 364
3.3 Epigraphs and extremal processes......Page 369
3.4 Increasing set-valued processes of excursion sets......Page 377
3.5 Strong law of large numbers for epigraphical sums......Page 379
3.6 Level sums of random upper semicontinuous functions......Page 382
3.7 Graphical convergence of random functions......Page 385
Notes to Chapter 5......Page 394
A  Topological spaces and linear spaces......Page 403
B  Space of closed sets......Page 414
C  Compact sets and the Hausdorff metric......Page 418
D  Multifunctions and semicontinuity......Page 425
E  Measures, probabilities and capacities......Page 428
F  Convex sets......Page 437
G  Semigroups and harmonic analysis......Page 441
H  Regular variation......Page 444
References......Page 450
List of Notation......Page 477
C......Page 481
G......Page 482
K......Page 483
M......Page 484
S......Page 485
W......Page 486
Z......Page 487
C......Page 488
E......Page 490
F......Page 491
I......Page 492
M......Page 493
N......Page 494
P......Page 495
R......Page 496
S......Page 498
U......Page 500
Z......Page 501