Stable probability measures on Euclidean spaces and on locally compact groups: structural properties and limit theorems
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht <<[u.a.]>>
Kluwer
2001
|
| Schriftenreihe: | Mathematics and its applications
531 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 612 S. |
| ISBN: | 1402000405 |
Internformat
MARC
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| 245 | 1 | 0 | |a Stable probability measures on Euclidean spaces and on locally compact groups |b structural properties and limit theorems |c by Wilfried Hazod and Eberhard Siebert |
| 264 | 1 | |a Dordrecht <<[u.a.]>> |b Kluwer |c 2001 | |
| 300 | |a XVII, 612 S. | ||
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| 490 | 1 | |a Mathematics and its applications |v 531 | |
| 700 | 1 | |a Siebert, Eberhard |e Verfasser |4 aut | |
| 830 | 0 | |a Mathematics and its applications |v 531 |w (DE-604)BV008163334 |9 531 | |
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Datensatz im Suchindex
| _version_ | 1804140505287622656 |
|---|---|
| adam_text | Titel: Stable probability measures on Euclidean spaces and on locally compact groups
Autor: Hazod, Wilfried
Jahr: 2001
Contents
Preface iii
Introduction xi
I Probabilities on vector spaces 1
§ 1.1 Preparations: Linear operators on finite-dimensional vector spaces . 3
I Notations (in particular for Chapter I)................ 4
II Discrete one-parameter groups of operators.............. 6
III Continuous one-parameter groups of operators............ 7
IV Linear groups.............................. 11
§ 1.2 F ill probability measures and convergence of types......... 11
§ 1.3 Operator-semistable laws and operator-stable laws.......... 17
I Definition and Lévy-Khinchin representation............. 17
II Annexe: More on infinitely divisible laws............... 25
§ 1.4 Levy measures of operator-(semi-) stable laws............ 26
I Levy measures of operator-semistable laws.............. 26
II Levy measures of operator-stable laws................. 29
§1.5 Algebraic characterization of operator-(semi-) stability....... 35
I The structure of Lin(ji)......................... 35
II Subordination and (semi-) stability.................. 41
III A randomized characterization of operator-stability......... 42
§ 1.6 Operator- (semi-) stable laws as limit distributions ......... 44
I Domains of operator- (semi-) attraction................ 44
II Annexe: More on limits of infinitely divisible laws.......... 49
III More on domains of operator semi-attraction............. 56
§ 1.7 Properties of operator- (semi-) stable laws.............. 62
§ 1.8 Exponents of operator-stable laws................... 67
§ 1.9 Elliptical symmetry and large symmetry groups........... 74
I Elliptically symmetric operator- (semi-) stable laws......... 74
II Large symmetry groups......................... 79
vi CONTENTS
§ 1.10 Domains of normal operator attraction................ 81
I Stable laws................................ 81
II Remarks on operator-semistable laws................. 90
III Moments and domains of attraction.................. 92
§ 1.11 The existence of commuting normalizations.............. 96
§ 1.12 More on the structure of the decomposability group Lin(p) ..... 100
I Semistability and strict semistability ................. 100
II Jordan decomposition and spectrum of normalizing operators . . . 106
III Marginal distributions of operator (semi-) stable laws........ 109
§ 1.13 More on convergence of types theorems................ 116
I Types and transformation groups................... 117
II Applications of the convergence of types theorem.......... 120
III Finite-dimensional vector spaces.................... 124
IV A method to construct full measures, given B ............ 125
V Some examples.............................. 129
VI Stochastic compactness and regular variation properties....... 133
§ 1.14 Probabilities with idempotent type. F-stable and completely stable
measures................................. 135
§ 1.15 Examples and counterexamples .................... 147
I Operator-stable laws on V = R2 and R3............... 147
II Subordination of stable laws...................... 150
III Probabilities with discrete symmetry group on V = R2........ 152
IV Marginal distributions of operator stable laws............ 158
V Convergence of types and idempotent types ............. 160
VI Limit laws and domains of attraction................. 164
VII Commuting normalizations....................... 170
§ 1.16 References and comments for Chapter I................ 171
II Probabilities on simply connected nilpotent Lie groups 181
§ 2.0 Probabilities on locally compact groups: Some fundamental theorems 183
I Continuous convolution semigroups and the structure of generating
functional................................ 183
II Convergence of continuous convolution semigroups.......... 188
III Discrete convolution semigroups.................... 194
IV Embedding theorems.......................... 195
V Annexe: Supports of convolution semigroups............. 199
§ 2.1 Probabilities on simply connected nilpotent Lie groups....... 200
I Discrete and continuous convolution semigroups: The translation
procedure................................. 200
II Automorphisms and contractible Lie groups. Some basic facts . . . 203
CONTENTS vii
III Some examples of contractible Lie groups............... 209
§ 2.2 Convergence of types and full measures................ 213
I Simply connected nilpotent Lie groups................ 213
II Some generalizations.......................... 220
§ 2.3 Semistable and stable continuous convolution semigroups on simply
connected nilpotent Lie groups..................... 223
§ 2.4 Levy measures of stable and semistable laws............. 231
I Levy measures of semistable laws................... 231
II Levy measures of stable laws...................... 233
§ 2.5 Algebraic characterization of (semi-) stability............. 235
I The structure of Lin(/i)......................... 235
II The structure of Inv(/j), resp. Inv(A)................. 240
III Subordination and semistability.................... 245
IV A randomized characterization of operator-stability......... 246
§ 2.6 (Semi-) stable laws as limit distributions............... 247
I Limit theorems and uniqueness of embedding for semistable laws . . 247
II Domains of (semi-) attraction..................... 256
§ 2.7 Properties of (semi-) stable laws.................... 263
I Absolute continuity and purity laws.................. 263
II Gaussian and Bochner stable measures................ 267
III Holomorphic convolution semigroups................. 271
IV Moments of (semi-) stable laws..................... 275
§ 2.8 Exponents of stable laws........................ 281
§ 2.9 Elliptical symmetry and large invariance groups........... 288
§ 2.10 Domains of normal attraction..................... 294
I Stable and semistable laws....................... 294
II Moments and domains of attraction.................. 300
§ 2.11 Probabilities with idempotent type: T-stable and completely stable
measures................................. 303
I Idempotent (infinitesimal) T-types and F-stable laws ........ 303
II Complete stability............................ 314
III Marginals and complete stability.................... 322
rV Intrinsic definitions of semistability.................. 326
§ 2.12 Domains of partial attraction and random limit theorems on groups
and vector spaces............................ 328
I The existence of universal laws (Doeblin laws)............ 328
II Stochastic compactness......................... 336
III Random limit theorems: Independent random times......... 337
§ 2.13 Geometric (semi-) stability....................... 354
I Geometric convolutions......................... 354
vüi CONTENTS
II Properties of geometric and exponential distributions........ 355
III Characterization of geometric convolutions and exponential mixtures 358
rV Geometric semistability......................... 363
V Geometric domains of attraction.................... 364
VI Illustrations and examples for vector spaces G=V......... 366
VII More arithmetic properties of geometric convolutions........ 368
§ 2.14 Remarks on self-decomposable laws on vector spaces and on groups 371
I The decomposability semigroup D(p)................. 371
II Self-decomposability........................... 375
III Cocycle equations, background driving processes and generalized
Ornstein-Uhlenbeck processes..................... 376
IV Stable hemigroups and self-similar processes............. 379
V Space-time processes.......................... 382
VI Processes on G and on V ....................... 383
VII Background driving processes with logarithmic moments...... 384
VIII Full self-decomposable distributions and limit laws.......... 390
IX Generalizations and examples..................... 392
§ 2.15 More limit theorems on G and V : Mixing properties and dependent
random times.............................. 398
I A theorem of H. Cramer........................ 399
II Limit theorems for mixing arrays of random variables........ 402
III Random limit theorems in the domain of attraction of (semi-) stable
laws: Dependent random times..................... 406
§ 2.16 References and comments for Chapter II............... 413
III (Semi-) stability and limit theorems on general locally compact
groups 427
§ 3.1 Contractive automorphisms on locally compact groups....... 428
I Contractive automorphisms and contractible groups......... 429
II Totally disconnected contractible groups............... 433
III The structure theorem for contractible groups............ 438
IV Contractive one-parameter automorphism groups.......... 440
V Some more structure theorems for discrete automorphism groups . 446
§ 3.2 Automorphisms contracting modulo a compact subgroup K .... 448
I Contraction mod K........................... 449
II The structure theorem: Ck{t) = C{r) K for discrete automorphism
groups acting on a Lie group...................... 454
III Borei cross-sections for the action of C(t) on Ck(j) (discrete auto-
morphism groups)............................ 458
IV Continuous automorphism groups................... 462
CONTENTS ix
V The structure theorem: Ck(T) = C(T) x K for continuous auto-
morphism groups............................465
VI The structure of Ck{t) for p-adic Lie groups............. 467
§ 3.3 Examples, counterexamples and some more structure theory .... 468
I Contractible and üf-contractible Lie groups.............. 468
II Automorphisms of compact groups .................. 475
III Infinite-dimensional tori and solenoidal groups............ 477
IV Retopologization of C(t): Intrinsic topologies of contractible groups 480
§ 3.4 (Semi-) stable convolution semigroups with trivial idempotents . . . 488
I General definitions of strictly (semi-) stable convolution semigroups 488
II (Semi-) stable continuous convolution semigroups with trivial idem-
potents .................................. 493
III Some examples and further remarks.................. 498
§ 3.5 (Semi-) stable convolution semigroups with nontrivial idempotents . 507
I Semistable convolution semigroups on Lie groups with nontrivial
idempotents............................... 509
II Stable convolution semigroups with nontrivial idempotents..... 512
III Semistable submonogeneous semigroups on Lie groups........ 514
IV Semistable convolution semigroups with nontrivial idempotents on
p-adic Lie groups ............................ 518
§ 3.6 More on probabilities on contractible groups............. 519
I Domains of partial attraction on contractible groups......... 519
II The existence of Doeblin laws on contractible groups........ 528
III A translation procedure for contractible locally compact groups . . 530
IV Point processes on groups and continuous convolution semigroups . 534
§ 3.7 Limit laws and convergence of types theorems. A survey...... 537
I Limits of discrete convolution semigroups with nontrivial idempotents 537
II Convergence of types theorems..................... 549
III Applications to semistability...................... 557
IV Limit laws on compact extensions of contractible groups JV x K . . 559
§ 3.8 References and comments for Chapter III............... 562
Epilogue 567
Bibliography 573
List of Symbols 601
Index 607
|
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| isbn | 1402000405 |
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| spelling | Hazod, Wilfried 1943-2014 Verfasser (DE-588)108801055 aut Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems by Wilfried Hazod and Eberhard Siebert Dordrecht <<[u.a.]>> Kluwer 2001 XVII, 612 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 531 Siebert, Eberhard Verfasser aut Mathematics and its applications 531 (DE-604)BV008163334 531 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018481552&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hazod, Wilfried 1943-2014 Siebert, Eberhard Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems Mathematics and its applications |
| title | Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems |
| title_auth | Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems |
| title_exact_search | Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems |
| title_full | Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems by Wilfried Hazod and Eberhard Siebert |
| title_fullStr | Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems by Wilfried Hazod and Eberhard Siebert |
| title_full_unstemmed | Stable probability measures on Euclidean spaces and on locally compact groups structural properties and limit theorems by Wilfried Hazod and Eberhard Siebert |
| title_short | Stable probability measures on Euclidean spaces and on locally compact groups |
| title_sort | stable probability measures on euclidean spaces and on locally compact groups structural properties and limit theorems |
| title_sub | structural properties and limit theorems |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018481552&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
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