Decomposition of random variables and vectors:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
1977
|
| Series: | Translations of mathematical monographs
48 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 380 S. |
| ISBN: | 0821815989 |
Staff View
MARC
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| 240 | 1 | 0 | |a Razloženija slučajnych veličin i vektorov |
| 245 | 1 | 0 | |a Decomposition of random variables and vectors |c by Ju. V. Linnik and I. V. Ostrovskij̆ |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1977 | |
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Record in the Search Index
| _version_ | 1804116419309207552 |
|---|---|
| adam_text | TABLE OF CONTENTS
Introduction 1
Chapter 1. Auxiliary material from probability theory and analysis 3
§ 1. Probability theory 3
§2. Functions of a complex variable 11
§3. Dini derivatives and their properties 15
§4. Elementary properties of continued fractions 16
§5. Elementary transformation formula for theta functions 17
§6. Simple examples of the method of steepest descent 18
Chapter II. Analytical properties of characteristic functions 21
§ 1. Differentiability conditions for characteristic functions 21
§2. Analyticity conditions for characteristic functions 23
§3. Ridge functions 28
§4. Entire characteristic functions 35
§5. Marcinkiewicz s theorem 41
§6. Analytic characteristic functions of infinitely divisible distributions... 45
Chapter III. General theorems on factorizations of probability distributions 55
§ 1. Components of distributions with analytic characteristic functions.... 56
§2. Spectrum of a distribution and the spectra of its components 61
§3. Indecomposable distributions 65
§4. Hincin s factorization theorem 78
§5. Description of the class/0 87
Chapter IV. Necessary conditions for membership of distributions with a
Gaussian component in Io 95
§1. Formulation of the fundamental theorem 95
§2. The three basic lemmas 96
§3. Use of steepest descent whenx 0 99
§4. Use of steepest descent when x is positive 100
§5. Transformations in the conditions of Lemma 1 101
§6. Investigation of U(t) 103
§7. Investigation of /lv(x) 104
§8. Completion of the proof of Lemma 1 106
§9. Use of continued fractions in the proof of Lemma II 108
§10. Investigation of U(t) under the assumptions of Lemma II 109
§11. The case |f| y/q, a0 A2 110
§12. Further investigation of the case a0 GA2 112
§13. The case a0 At 113
vii
TABLE OF CONTENTS
§14. Behavior of U(t) when a0 GAt 114
§15. Completion of the proof of Lemma II 115
§16. Use of Dini derivatives in the proof of Lemma HI 116
§17. The intervals 7rx and 7r(S0) 117
§18. Small values of U{t) under the assumptions of Lemma III 119
§19. Case of rational a in Lemma III 120
§20. Case of rational a, concluded 124
§21. Case of irrational a 125
§22. Completion of the proof of Lemma III 126
§23. Proof of Theorem 4.1.1 using the three basic lemmas 128
Chapter V. Sufficient conditions for membership of distributions in the
class Q in /„ 131
§ 1. Theorem on decomposition of convolutions of Gaussian and Pois
son distributions 131
§2. Some auxiliary results from the theory of analytic functions 138
§3. Proof of the main theorem 145
§4. Lattice distributions in the class $3 160
§5. Example of a distribution in CVq 165
Chapter VI. Factorizations of multi dimensional distributions 169
§1. Projection method. Analytic characteristic functions 170
§2. General theorems on factorizations of multidimensional distribu¬
tions 182
§3. Factorizations of convolutions of multidimensional Gaussian and
Poisson distributions 190
§4. Sufficient conditions for membership of infinitely divisible dis¬
tributions with no Gaussian component in IOn 195
§5. Auxiliary results 198
§6. Proof of Theorems 6.4.1, 6.4.2 and 6.4.3 208
§7. Sufficient conditions for nonmembership of infinitely divisible
distributions in the class IOn 218
Chapter VII. Generalization of the factorization problem 227
§ 1. a factorization 227
§2. Auxiliary theorems on finite a factorizations 228
§3. Auxiliary theorems on countable a factorizations 234
§4. Finite and countable a factorizations of a normal distribution. Ap¬
plications 249
§5. Infinitely divisible distributions having only infinitely divisible a
components 253
Chapter VIII. Stability of factorizations 257
§1. General stability theorem for factorizations 257
viii
TABLE OF CONTENTS
§2. Sapogov s theorem 260
§3. On the sharpness of Sapogov s theorem 266
§4. Stability estimate for Cramer s theorem in the Levy metric 270
§5. Stability estimate for Ralkov s theorem 274
§6. Survey of other results 282
Chapter IX. Limit theorems without the asymptotic negligibility condition 285
§ 1. The classical approach to the addition of independent random vari¬
ables 285
§2. The nonclassical formulation of the problem of addition of inde¬
pendent random variables 288
§3. Generalization of the Lindeberg Feller theorem 289
§4. Convergence to a Poisson distribution 296
§5. Convergence to distributions in the class (} 301
§6. Generalization to random variables with values in a Hilbert space 304
Chapter X. Unsolved problems 307
Chapter II 307
Chapter III 308
Chapter IV 308
Chapter V 308
Chapter VI 309
Chapter VII 310
Chapters VIII and IX 311
Appendix I. Proof of the Levy Hintin formula using the Krem Mil man
theorem 313
§1. The Krem Mil man theorem and its simplest applications 313
§2. Proof of the Iivy Hincin formula 317
Appendix II. A certain class of characteristic function 335
Appendix III. On factorizations of distributions as convolutions of functions
of bounded variation 349
Appendix IV. On Gaussian components of multidimensional probability
distributions 355
Notes 359
Bibliography 369
Subject index 378
Author s index 381
ix
|
| any_adam_object | 1 |
| author | Linnik, Jurij V. 1915-1972 Ostrovskij, Josif V. |
| author_GND | (DE-588)117715247 |
| author_facet | Linnik, Jurij V. 1915-1972 Ostrovskij, Josif V. |
| author_role | aut aut |
| author_sort | Linnik, Jurij V. 1915-1972 |
| author_variant | j v l jv jvl j v o jv jvo |
| building | Verbundindex |
| bvnumber | BV001974909 |
| classification_rvk | SK 800 |
| ctrlnum | (OCoLC)251793539 (DE-599)BVBBV001974909 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV001974909 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:38:12Z |
| institution | BVB |
| isbn | 0821815989 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001288124 |
| oclc_num | 251793539 |
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| physical | IX, 380 S. |
| psigel | TUB-nveb |
| publishDate | 1977 |
| publishDateSearch | 1977 |
| publishDateSort | 1977 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Linnik, Jurij V. 1915-1972 Verfasser (DE-588)117715247 aut Razloženija slučajnych veličin i vektorov Decomposition of random variables and vectors by Ju. V. Linnik and I. V. Ostrovskij̆ Providence, RI American Math. Soc. 1977 IX, 380 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 48 Zufallsvariable (DE-588)4129514-6 gnd rswk-swf Faktorenanalyse (DE-588)4016338-6 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Zufallsvariable (DE-588)4129514-6 s Faktorenanalyse (DE-588)4016338-6 s DE-604 Wahrscheinlichkeitstheorie (DE-588)4079013-7 s 1\p DE-604 Ostrovskij, Josif V. Verfasser aut Translations of mathematical monographs 48 (DE-604)BV000002394 48 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001288124&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Linnik, Jurij V. 1915-1972 Ostrovskij, Josif V. Decomposition of random variables and vectors Translations of mathematical monographs Zufallsvariable (DE-588)4129514-6 gnd Faktorenanalyse (DE-588)4016338-6 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4129514-6 (DE-588)4016338-6 (DE-588)4079013-7 |
| title | Decomposition of random variables and vectors |
| title_alt | Razloženija slučajnych veličin i vektorov |
| title_auth | Decomposition of random variables and vectors |
| title_exact_search | Decomposition of random variables and vectors |
| title_full | Decomposition of random variables and vectors by Ju. V. Linnik and I. V. Ostrovskij̆ |
| title_fullStr | Decomposition of random variables and vectors by Ju. V. Linnik and I. V. Ostrovskij̆ |
| title_full_unstemmed | Decomposition of random variables and vectors by Ju. V. Linnik and I. V. Ostrovskij̆ |
| title_short | Decomposition of random variables and vectors |
| title_sort | decomposition of random variables and vectors |
| topic | Zufallsvariable (DE-588)4129514-6 gnd Faktorenanalyse (DE-588)4016338-6 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Zufallsvariable Faktorenanalyse Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001288124&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
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