Independent and stationary sequences of random variables:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Groningen
Wolters-Noordhoff
1971
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 443 S. |
| ISBN: | 9001418856 |
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MARC
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| 245 | 1 | 0 | |a Independent and stationary sequences of random variables |c I. A. Ibragimov and Yu. V. Linnik. Ed. by J. F. C. Kingman |
| 264 | 1 | |a Groningen |b Wolters-Noordhoff |c 1971 | |
| 300 | |a 443 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Distribution (Théorie des probabilités) | |
| 650 | 4 | |a Processus stables | |
| 650 | 7 | |a Stochastische variabelen |2 gtt | |
| 650 | 4 | |a Suites (Mathématiques) | |
| 650 | 4 | |a Distribution (Probability theory) | |
| 650 | 4 | |a Random variables | |
| 650 | 4 | |a Sequences (Mathematics) | |
| 650 | 4 | |a Stationary sequences (Mathematics) | |
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Record in the Search Index
| _version_ | 1804117098563108864 |
|---|---|
| adam_text | CONTENTS
Editor s note
Preface _
Chapter 1
Probability distributions on the real line: infinitely divisible laws 17
1. Probability spaces, conditional probabilities and expectations 17
2. Distributions and distribution functions 19
3. Convergence of distributions 21
4. Moments and characteristic functions 24
5. Continuity of the correspondence between distributions and
characteristic functions 27
6. A special theorem about characteristic functions 32
7. Infinitely divisible distributions 34
Chapter 2
Stable distributions; analytical properties and domains of attraction 37
1. Stable distributions 37
2. Canonical representation of stable laws 39
3. Analytic structure of the densities of stable distributions ... 47
4. Asymptotic formulae for the densities p(x; a, #) 54
5. Unimodality of stable laws 66
6. Domains of attraction 76
6 CONTENTS
Chapter 3
Refinements of the limit theorems for normal convergence 94
1. Introduction 94
2. Some auxiliary theorems 94
3. The deviation Rn(x) 97
4. Necessary and sufficient conditions 104
5. The maximum deviation of Fn from 0 Ill
6. Dependence of the remainder term on n and x 117
Chapter 4
Local limit theorems 120
1. Formulation of the problem 120
2. Local limit theorems for lattice distributions 121
3. A limit theorem for densities 125
4. Limit theorems in the Lt metric 128
5. A refinement of the local limit theorems for the case of normal
convergence 135
Chapter 5
Limit theorems in Lp spaces 139
1. Statement of the problem 139
2. Domains of attraction of stable laws in the Lp metric .... 141
3. Estimates of || Fn — [ p in the case of normal convergence ... 146
Chapter 6
Limit theorems for large deviations 154
1. Introduction and examples 154
2. Statement of the problem 158
CONTENTS 7
Chapter 7
Richter s local theorems and Bernstein s inequality 160
1. Statement of the theorems 160
2. A local limit theorem for probability densities 161
3. Calculation of the integral near a saddle point 166
4. A local limit theorem for lattice variables 167
5. Bernstein s inequality 169
Chapter 8
Cramer s integral theorem and its refinement by Petrov 171
1. Statement of the theorem 171
2. The introduction of auxiliary random variables 172
3. Proof of the theorem 174
Chapter 9
Monomial zones of local normal attraction 177
1. Zones of normal attraction 177
2. The fundamental conditions . 178
3. Fundamental theorems 180
4. Approximation of the characteristic function by a finite Taylor
series 182
5. Derivation of the basic integral 184
6. Completion of the proof 187
Chapter 10
Monomial zonesof local attraction to Cramer s system of limiting tails 190
1. Formulation 190
2. On the condition (10.1.9) 192
8 CONTENTS
3. Derivation of the fundamental integral 192
4. Application of the method of steepest descents 194
5. Completion of the proof of Theorem 10.1.1 197
Chapter 11
Narrow zones of normal attraction 198
1. Classification of narrow zones by the function h 198
2. Statement of the theorems r. . . . 199
3. On the conditions imposed upon h(x) 200
4. The necessity of (11.2.2) for Class 1 200
5. The sufficiency of (11.2.2) for Class I 201
6. Investigation of the fundamental integral 203
7. More investigation of the fundamental integral 204
8. Investigation of K{t) 207
9. More investigation of K{t) 209
10. Completion of the proof of Theorem 11.2.1 211
11. The corresponding integral theorem 212
12. Calculation of the auxiliary limit distribution 214
13. More about the auxiliary limit distribution 215
14. Completion of the proof of Theorem 11.2.2 217
15. The general case of narrow zones 218
16. The transition to Theorems 11.2.3 5 220
17. Choice of /j, 222
18. Completion of the proof 224
Chapter 12
Wide monomial zones of integral normal attraction 226
1. Formulation 226
2. An upper bound for the probability of a large deviation . . . 227
3. Introduction of auxiliary variables 229
4. Study of the basic relation 231
5. Derivation of the fundamental formula 232
CONTENTS 9
6. The fundamental integral formula 234
7. Study of the auxiliary integral 235
8. Expansion of R as a Taylor series 236
9. Further transformations 238
10. Completion of the proof of sufficiency 240
11. Proof of the necessity 241
12. Completion of the proof 243
Chapter 13
Monomial zones of integral attraction to Cramer s system of limiting
tails 244
1. Formulation 244
2. An upper bound for the probability of a large derivation . . . 245
3. Investigation of the basic formula 251
4. Completion of the proof 253
Chapter 14
Integral theorems holding on the whole line 254
1. Formulation 254
2. An elementary result on the probability of very large deviations 255
3. Radial extensions 258
4. Investigation of the fundamental integral 260
5. Investigation of the auxiliary integrals 263
6. An example 265
Chapter 15
Approximation of distributions of sums of independent components
by infinitely divisible distributions 267
1. Statement of the problem 267
2. Concentration functions 268
10 CONTENTS
3. Auxiliary propositions 273
4. Proof of Theorem 15.1.1 278
Chapter 16
Some results from the theory of stationary processes 284
1. Definition and general properties 284
2. Stationary processes and the associated measure preserving
transformations 286
3. Hilbert spaces associated with a stationary process 288
4. Autocovariance and spectral functions of stationary processes 291
5. The spectral representation of stationary processes 292
6. The structure of Lx and linear transformations of stationary
processes 296
7. Existence theorems for the spectral density 298
Chapter 17
Conditions of weak dependence for stationary processes 301
1. Regularity 301
2. The strong mixing condition 305
3. Conditions of weak dependence for Gaussian sequences ... 310
Chapter 18
The central limit theorem for stationary processes 315
1. Statement of the problem 315
2. The variance of Xl+...+Xn 321
3. The variance of the integral $1 X(t)dt 330
4. The central limit theorem for strongly mixing sequences ... 333
5. Sufficient conditions for the central limit theorem 340
CONTENTS 11
6. The central limit theorem for functional of mixing sequences 352
7. The central limit theorem in continuous time 362
Chapter 19
Examples and addenda 365
1. The central limit theorem for homogeneous Markov chains. . 365
2. m dependent sequences 369
3. The distribution of values of sums of the form 2/(2* x). ... 370
4. Application to the metric theory of continued fractions . . . 374
5. Example of a sequence not satisfying the central limit theorem 384
Chapter 20
Some unsolved problems 390
Appendix 1
Sowly varying functions 394
Appendix 2
Theorems on Fourier transforms 398
Appendix 3
A theorem on convergence of conditional expectations 400
Notes 401
Some contributions of recent years 406
by I. A. Ibragimov, V. V. Petrov
Bibliography 429
Subject index 440
|
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| author | Ibragimov, Ilʹdar Abdullovič 1932- Linnik, Jurij V. 1915-1972 |
| author_GND | (DE-588)172161991 (DE-588)117715247 (DE-588)108793877 |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:49:00Z |
| institution | BVB |
| isbn | 9001418856 |
| language | English |
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| physical | 443 S. |
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| spelling | Ibragimov, Ilʹdar Abdullovič 1932- Verfasser (DE-588)172161991 aut Independent and stationary sequences of random variables I. A. Ibragimov and Yu. V. Linnik. Ed. by J. F. C. Kingman Groningen Wolters-Noordhoff 1971 443 S. txt rdacontent n rdamedia nc rdacarrier Distribution (Théorie des probabilités) Processus stables Stochastische variabelen gtt Suites (Mathématiques) Distribution (Probability theory) Random variables Sequences (Mathematics) Stationary sequences (Mathematics) Zufallsvariable (DE-588)4129514-6 gnd rswk-swf Zufällige Folge (DE-588)4191092-8 gnd rswk-swf Zufallsvariable (DE-588)4129514-6 s DE-604 Zufällige Folge (DE-588)4191092-8 s Linnik, Jurij V. 1915-1972 Verfasser (DE-588)117715247 aut Kingman, J. F. C. 1939- Sonstige (DE-588)108793877 oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001755338&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ibragimov, Ilʹdar Abdullovič 1932- Linnik, Jurij V. 1915-1972 Independent and stationary sequences of random variables Distribution (Théorie des probabilités) Processus stables Stochastische variabelen gtt Suites (Mathématiques) Distribution (Probability theory) Random variables Sequences (Mathematics) Stationary sequences (Mathematics) Zufallsvariable (DE-588)4129514-6 gnd Zufällige Folge (DE-588)4191092-8 gnd |
| subject_GND | (DE-588)4129514-6 (DE-588)4191092-8 |
| title | Independent and stationary sequences of random variables |
| title_auth | Independent and stationary sequences of random variables |
| title_exact_search | Independent and stationary sequences of random variables |
| title_full | Independent and stationary sequences of random variables I. A. Ibragimov and Yu. V. Linnik. Ed. by J. F. C. Kingman |
| title_fullStr | Independent and stationary sequences of random variables I. A. Ibragimov and Yu. V. Linnik. Ed. by J. F. C. Kingman |
| title_full_unstemmed | Independent and stationary sequences of random variables I. A. Ibragimov and Yu. V. Linnik. Ed. by J. F. C. Kingman |
| title_short | Independent and stationary sequences of random variables |
| title_sort | independent and stationary sequences of random variables |
| topic | Distribution (Théorie des probabilités) Processus stables Stochastische variabelen gtt Suites (Mathématiques) Distribution (Probability theory) Random variables Sequences (Mathematics) Stationary sequences (Mathematics) Zufallsvariable (DE-588)4129514-6 gnd Zufällige Folge (DE-588)4191092-8 gnd |
| topic_facet | Distribution (Théorie des probabilités) Processus stables Stochastische variabelen Suites (Mathématiques) Distribution (Probability theory) Random variables Sequences (Mathematics) Stationary sequences (Mathematics) Zufallsvariable Zufällige Folge |
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