A modern approach to probability theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
1997
|
| Series: | Probability and its applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 756 Seiten Illustrationen |
| ISBN: | 0817638075 3764338075 |
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| 245 | 1 | 0 | |a A modern approach to probability theory |c Bert Fristedt ; Lawrence Gray |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1997 | |
| 300 | |a XX, 756 Seiten |b Illustrationen | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
List of Tables xv
Preface xvi
Part I. Probability Spaces, Random Variables, and Expectations
Chapter 1. Probability Spaces 3
1.1. Introductory examples 3
1.2. Ingredients of probability spaces 6
1.3. cr fields 8
1.4. Borel tr nelds 9
Chapter 2. Random Variables 11
2.1. Definitions and basic results 11
2.2. Revalued random variables 14
2.3. K°° valued random variables 18
2.4. Further examples 20
Chapter 3. Distribution Functions 25
3.1. Basic theory 25
3.2. Examples of distributions 29
3.3. Some descriptive terminology 31
3.4. Distributions with densities 35
3.5. Further examples 37
3.6. Distribution functions for the extended real line 39
vi CONTENTS
Chapter 4. Expectations: Theory 41
4.1. Definitions 41
4.2. Linearity and positivity 46
4.3. Monotone convergence 49
4.4. Expectation of compositions 51
4.5. The Riemann Stieltjes integral and expectations 53
Chapter 5. Expectations: Applications 59
5.1. Variance and the Law of Large Numbers 59
5.2. Mean vectors and covariance matrices 66
5.3. Moments and the Jensen Inequality 68
5.4. Probability generating functions 70
5.5. Characterization of probability generating functions 73
Chapter 6. Calculating Probabilities and Measures 75
6.1. Operations on events 75
6.2. The Borel Cantelli and Kochen Stone Lemmas 77
6.3. Inclusion exclusion 80
6.4. Finite and tr nnite measures 81
Chapter 7. Measure Theory: Existence and Uniqueness 85
7.1. The Sierpinski Class Theorem and uniqueness 85
7.2. Finitely additive functions defined on fields 87
7.3. Existence, extension, and completion of measures 90
7.4. Examples 95
Chapter 8. Integration Theory 101
8.1. Lebesgue integration 101
8.2. Convergence theorems 105
8.3. Probability measures and infinite measures compared 110
8.4. Lebesgue integrals and Riemann Stieltjes integrals Ill
8.5. Absolute continuity and densities 115
8.6. Integration with respect to counting measure 118
CONTENTS vii
Part 2. Independence and Sums
Chapter 9. Stochastic Independence 121
9.1. Definition and basic properties 121
9.2. Product measure: finitely many factors 127
9.3. The Fubini Theorem 130
9.4. Expectations and independence 133
9.5. Densities and independence 134
9.6. Product probability measure: infinitely many factors 136
9.7. The Borel Cantelli Lemma and independent sequences 141
9.8. f Order statistics 143
9.9. f Some new distributions involving independence 145
Chapter 10. Sums of Independent Random Variables 147
10.1. Convolutions of distributions 147
10.2. Multinomial distributions 152
10.3. Probability generating functions and sums in Z 153
10.4. t Dirichlet distributions 156
10.5. f Random sums in various settings 159
Chapter 11. Random Walk 163
11.1. Random sequences 163
11.2. Definition and examples 164
11.3. Filtrations and stopping times 171
11.4. Stopping times and random walks 174
11.5. A hitting time example 175
11.6. Returns to 0 178
11.7. f Random walks in various settings 183
Chapter 12. Theorems of A.S. Convergence 185
12.1. Convergence in probability 185
12.2. Laws of Large Numbers 188
12.3. Applications 193
12.4. 0 1 laws 196
12.5. Random infinite series 198
viii CONTENTS
12.6. The Etemadi Lemma 200
12.7. f The Kolmogorov Three Series Theorem 202
12.8. f The image of a random walk 206
Chapter 13. Characteristic Functions 209
13.1. Definition and basic examples 209
13.2. The Parseval Relation and uniqueness 211
13.3. Characteristic functions of convolutions 214
13.4. Symmetrization 217
13.5. Moment generating functions 218
13.6. Moment theorems 223
13.7. Inversion theorems 226
13.8. Characteristic functions in Rd 234
13.9. Normal distributions on d dimensional space 237
13.10. f An application to random walks on Z 238
13.11. f An application to the calculation of a sum 239
Part 3. Convergence in Distribution
Chapter 14. Convergence in Distribution on the Real Line 243
14.1. Definitions and examples 243
14.2. Limit distributions for extreme values 246
14.3. Relationships to other types of convergence 249
14.4. Convergence conditions for sequences of distributions 252
14.5. Sequences of distributions on R 254
14.6. Relative sequential compactness 255
14.7. The Continuity Theorem 260
14.8. Scaling and centering of sequences of distributions 263
14.9. Characterization of moment generating functions 266
14.10. Characterization of characteristic functions 269
Chapter 15. Distributional Limit Theorems for Partial Sums 271
15.1. Infinite series of independent random variables 272
15.2. The Law of Large Numbers revisited 273
15.3. The Classical Central Limit Theorem 275
CONTENTS ix
15.4. The general setting for iid sequences 277
15.5. f Large deviations 280
15.6. f Local limit theorems 282
Chapter 16. Infinitely Divisible Distributions as Limits 289
16.1. Compound Poisson distributions 289
16.2. Infinitely divisible distributions on R 293
16.3. Levy Khinchin representations 295
16.4. Infinitely divisible distributions onR+ 300
16.5. Extension to S+ 302
16.6. The triangular array problem: introduction 303
16.7. lid triangular arrays 306
16.8. Symmetric and nonnegative triangular arrays 310
16.9. f General triangular arrays 313
Chapter 17. Stable Distributions as Limits 323
17.1. Regular variation 324
17.2. The stable distributions 326
17.3. f Domains of attraction 332
17.4. % Domains of strict attraction 342
Chapter 18. Convergence in Distribution on Polish Spaces 347
18.1. Polish spaces 347
18.2. Definition of and criteria for convergence 352
18.3. Relative sequential compactness 355
18.4. Uniform tightness and the Prohorov Theorem 357
18.5. Convergence in product spaces 358
18.6. The Continuity Theorem for Ed 361
18.7. f The Prohorov metric 363
Chapter 19. The Invariance Principle and Brownian Motion 367
19.1. Certain sequences of distributions on C[0,1] 368
19.2. The existence of and convergence to Wiener measure 371
19.3. Some measurable functional on C[0,1] 374
19.4. Brownian motion on [0, oo) 379
x CONTENTS
19.5. Filtrations and stopping times 382
19.6. Brownian motion, nitrations, and stopping times 385
19.7. X Characterization of Brownian motion 389
19.8. f Law of the Iterated Logarithm 390
Part 4. Conditioning
Chapter 20. Spaces of Random Variables 395
20.1. Hilbert spaces 395
20.2. The Hilbert space L2(fi, F,P) 397
20.3. The metric space Li(O, F, P) 401
20.4. f Best linear estimator 402
Chapter 21. Conditional Probabilities 403
21.1. The construction of conditional probabilities 403
21.2. Conditional distributions 412
21.3. Conditional densities 416
21.4. Existence and uniqueness of conditional distributions 417
21.5. Conditional independence 422
21.6. | Conditional distributions of normal random vectors 427
Chapter 22. Construction of Random Sequences 429
22.1. The basic result 429
22.2. Construction of exchangeable sequences 433
22.3. Construction of Markov sequences 436
22.4. Polya urns 437
22.5. f Coupon collecting 439
Chapter 23. Conditional Expectations 443
23.1. Definition of conditional expectation 443
23.2. Conditional versions of unconditional theorems 448
23.3. Formulas for conditional expectations 451
23.4. Conditional variance 453
CONTENTS xi
Part 5. Random Sequences
Chapter 24. Martingales 459
24.1. Basic definitions 459
24.2. Examples 461
24.3. Doob decomposition 465
24.4. Transformations of submartingales 466
24.5. Another transformation: optional sampling 467
24.6. Applications of optional sampling 473
24.7. Inequalities and convergence results 477
24.8. f Optimal strategy in Red and Black 484
Chapter 25. Renewal Sequences 489
25.1. Basic criterion 490
25.2. Renewal measures and potential measures 491
25.3. Examples 494
25.4. Renewal theory: a first step 497
25.5. Delayed renewal sequences 499
25.6. The Renewal Theorem 502
25.7. f Applications to random walks 506
Chapter 26. Time homogeneous Markov Sequences 511
26.1. Transition operators and discrete generators 511
26.2. Examples 515
26.3. Martingales and the strong Markov property 520
26.4. Hitting times and return times 522
26.5. Renewal theory and Markov sequences 525
26.6. Irreducible Markov sequences 527
26.7. Equilibrium distributions 528
Chapter 27. Exchangeable Sequences 533
27.1. Finite exchangeable sequences 533
27.2. Infinite exchangeable sequences 539
27.3. Posterior distributions 542
27.4. f Generalization to Borel spaces 544
xii CONTENTS
27.5. X Ferguson distributions and Blackwell MacQueen urns 550
Chapter 28. Stationary Sequences 553
28.1. Definitions 553
28.2. Notation 555
28.3. Examples 556
28.4. The Birkhoff Ergodic Theorem 558
28.5. Ergodicity 561
28.6. f The Kingman Liggett Subadditive Ergodic Theorem 564
28.7. % Spectral analysis of stationary sequences 571
Part 6. Stochastic Processes
Chapter 29. Point Processes 581
29.1. Point processes as random Radon measures 581
29.2. Intensity measures 586
29.3. Poisson point processes 587
29.4. Examples of Poisson point processes 589
29.5. f Probability generating functionals 594
29.6. X Operations on point processes 597
29.7. | Convergence in distribution for point processes 598
Chapter 30. Levy Processes 601
30.1. Measurable spaces of right continuous functions 601
30.2. Definition of Levy process 602
30.3. Construction of Levy processes 605
30.4. Filtrations and stopping times 610
30.5. f Subordination 611
30.6. X Local time processes and regenerative subsets of [0, oo) 612
30.7. X Sample function properties of subordinators 618
Chapter 31. Introduction to Markov Processes 621
31.1. Cadlag space 621
31.2. Markov, strong Markov, and Feller processes 622
31.3. Infinitesimal generators 628
31.4. The martingale problem 629
CONTENTS xiii
31.5. Pure jump Markov processes: bounded rates 632
31.6. Pure jump Markov processes: unbounded rates 636
31.7. | Renewal theory for pure jump Markov processes 639
Chapter 32. Interacting particle systems 641
32.1. Configuration spaces and infinitesimal generators 641
32.2. The universal coupling 644
32.3. Examples 651
32.4. Equilibrium distributions 655
32.5. Systems with attractive infinitesimal generators 657
Chapter 33. Diffusions and Stochastic Calculus 661
33.1. Stochastic difference equations 661
33.2. The Ito integral 663
33.3. Stochastic differentials and the Ito Lemma 668
33.4. Autonomous stochastic differential equations 672
33.5. Generators and the Dirichlet problem 679
33.6. Diffusions in higher dimensions 682
Part 7. Appendices
Appendix A. Notation and Usage of Terms 687
A.I. Symbols 687
A.2. Usage 690
A.3. Exercises on subtle distinctions 692
Appendix B. Metric Spaces 693
B.I. Definition 693
B.2. Sequences 694
B.3. Continuous functions 695
B.4. Important metric spaces 695
Appendix C. Topological Spaces 697
C.I. Concepts 697
C.2. Compactification 699
C.3. Product topologies 700
C.4. Relative topology 700
xiv CONTENTS
C.5. Limits and continuous functions 701
Appendix D. Riemann Stieltjes Integration 703
D.I. The Riemann Stieltjes integral 703
D.2. Relation to the Riemann integral 705
D.3. Change of variables 706
D.4. Integration by parts 707
D.5. Improper Riemann Stieltjes integrals 708
Appendix E. Taylor Approximations, C Valued Logarithms 709
E.I. Some inequalities based on the Taylor formula 709
E.2. Complex exponentials and logarithms 711
E.3. Approximations of general C valued functions 714
Appendix F. Bibliography 715
Appendix G. Comments and Credits 723
Index 737
|
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| isbn | 0817638075 3764338075 |
| language | English |
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| spelling | Fristedt, Bert 1937- Verfasser (DE-588)1145590519 aut A modern approach to probability theory Bert Fristedt ; Lawrence Gray Boston [u.a.] Birkhäuser 1997 XX, 756 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Probability and its applications Probabilités ram Waarschijnlijkheidstheorie gtt Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Gray, Lawrence F. 1949- Verfasser (DE-588)17211750X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007487298&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fristedt, Bert 1937- Gray, Lawrence F. 1949- A modern approach to probability theory Probabilités ram Waarschijnlijkheidstheorie gtt Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4079013-7 |
| title | A modern approach to probability theory |
| title_auth | A modern approach to probability theory |
| title_exact_search | A modern approach to probability theory |
| title_full | A modern approach to probability theory Bert Fristedt ; Lawrence Gray |
| title_fullStr | A modern approach to probability theory Bert Fristedt ; Lawrence Gray |
| title_full_unstemmed | A modern approach to probability theory Bert Fristedt ; Lawrence Gray |
| title_short | A modern approach to probability theory |
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| topic | Probabilités ram Waarschijnlijkheidstheorie gtt Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilités Waarschijnlijkheidstheorie Probabilities Wahrscheinlichkeitstheorie |
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