Advanced probability theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
1988
|
| Edition: | 1. print. |
| Series: | Probability, pure and applied
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 405 S. |
| ISBN: | 0824778731 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface iii
1. REVIEW OF BASIC CONCEPTS 1
1.1 Events 1
1.2 Probability 5
1.3 A method of constructing probabilities 12
1.4 Classical probability spaces 19
1.5 Simple random variables 20
1.6 Conditional probability: The elementary case 26
1.7 Independence of events 28
1.8 Random variables, distribution functions 33
References 38
2. EXPECTATION AND INTEGRAL, WEAK AND STRONG
CONVERGENCE 41
2.1 Definitions of expectation and integral 41
2.2 Basic properties of expectation 45
2.3 The inequalities of Markov, Chebyshev, and Kolmogorov 48
2.4 Sequences of integrals 54
2.5 The strong law of large numbers: The proof of Theorem 1 57
2.6 Zero one laws 62
2.7 Lebesgue Stieltjes integrals 66
2.8 Weak convergence 76
References 82
v
v; Contents
3. TRANSFORMS OF DISTRIBUTION 85
3.1 Characteristic functions: Basic properties 85
3.2 Characteristic functions: The uniqueness and continuity
theorems 93
3.3 Classical forms of the central limit theorem, and a model
for measurement errors 106
3.4 Characteristic functions: Inequalities 115
3.5 Multivariate characteristic functions 122
3.6 Laplace transforms 124
3.7 Generating functions 128
References 132
4. INFINITE SEQUENCES OF INDEPENDENT RANDOM
VARIABLES: WEAK CONVERGENCE 135
4.1 Complete convergence of sums without normalization,
infinite series 135
4.2 Decomposition of the normal distribution 146
4.3 Levy s metric 150
4.4 Zolotarev s theorem on asymptotic normality, the theorem
of Lindeberg and Feller 157
4.5 Speed of convergence: The Berry Esseen theorem 165
4.6 The class L of limiting distributions 171
References 182
5. TRIANGULAR ARRAYS OF INDEPENDENT RANDOM
VARIABLES, INFINITELY DIVISIBLE DISTRIBUTIONS 185
5.1 Introduction 185
5.2 More on infinitely divisible distributions 189
5.3 Convergence under UAN 196
5.4 Convergence to special distributions 204
References 216
6. INDEPENDENT AND IDENTICALLY DISTRIBUTED
RANDOM VARIABLES 219
6.1 Cauchy s functional equation 219
6.2 Stable distributions 224
6.3 Regularly varying functions 232
6.4 Domains of attraction for stable distributions 238
6.5 The asymptotic theory of the extremes 244
Contents vii
6.6 The law of the iterated logarithm 253
References 260
7. CONDITIONAL EXPECTATION; MARTINGALES 261
7.1 Conditional expectation, given a discrete random variable 261
7.2 Radon Nikodym theorem 266
7.3 Conditional expectation: The general case 271
7.4 Martingales 281
7.5 LP spaces 291
7.6 Further limit theorems for martingales 297
7.7 Exchangeability, De Finetti s theorem 306
References 312
8. TOPICS IN THE THEORY OF STOCHASTIC PROCESSES 315
8.1 Foundations and basic concepts 315
8.2 Poisson process 321
8.3 Renewal processes 328
8.4 The Galton Watson process; busy periods in queues 336
8.5 Markov chains 341
8.6 An invariance principle 353
References 359
Hints for Solutions of Exercises 361
Index 397
|
| adam_txt |
Contents
Preface iii
1. REVIEW OF BASIC CONCEPTS 1
1.1 Events 1
1.2 Probability 5
1.3 A method of constructing probabilities 12
1.4 Classical probability spaces 19
1.5 Simple random variables 20
1.6 Conditional probability: The elementary case 26
1.7 Independence of events 28
1.8 Random variables, distribution functions 33
References 38
2. EXPECTATION AND INTEGRAL, WEAK AND STRONG
CONVERGENCE 41
2.1 Definitions of expectation and integral 41
2.2 Basic properties of expectation 45
2.3 The inequalities of Markov, Chebyshev, and Kolmogorov 48
2.4 Sequences of integrals 54
2.5 The strong law of large numbers: The proof of Theorem 1 57
2.6 Zero one laws 62
2.7 Lebesgue Stieltjes integrals 66
2.8 Weak convergence 76
References 82
v
v; Contents
3. TRANSFORMS OF DISTRIBUTION 85
3.1 Characteristic functions: Basic properties 85
3.2 Characteristic functions: The uniqueness and continuity
theorems 93
3.3 Classical forms of the central limit theorem, and a model
for measurement errors 106
3.4 Characteristic functions: Inequalities 115
3.5 Multivariate characteristic functions 122
3.6 Laplace transforms 124
3.7 Generating functions 128
References 132
4. INFINITE SEQUENCES OF INDEPENDENT RANDOM
VARIABLES: WEAK CONVERGENCE 135
4.1 Complete convergence of sums without normalization,
infinite series 135
4.2 Decomposition of the normal distribution 146
4.3 Levy's metric 150
4.4 Zolotarev's theorem on asymptotic normality, the theorem
of Lindeberg and Feller 157
4.5 Speed of convergence: The Berry Esseen theorem 165
4.6 The class L of limiting distributions 171
References 182
5. TRIANGULAR ARRAYS OF INDEPENDENT RANDOM
VARIABLES, INFINITELY DIVISIBLE DISTRIBUTIONS 185
5.1 Introduction 185
5.2 More on infinitely divisible distributions 189
5.3 Convergence under UAN 196
5.4 Convergence to special distributions 204
References 216
6. INDEPENDENT AND IDENTICALLY DISTRIBUTED
RANDOM VARIABLES 219
6.1 Cauchy's functional equation 219
6.2 Stable distributions 224
6.3 Regularly varying functions 232
6.4 Domains of attraction for stable distributions 238
6.5 The asymptotic theory of the extremes 244
Contents vii
6.6 The law of the iterated logarithm 253
References 260
7. CONDITIONAL EXPECTATION; MARTINGALES 261
7.1 Conditional expectation, given a discrete random variable 261
7.2 Radon Nikodym theorem 266
7.3 Conditional expectation: The general case 271
7.4 Martingales 281
7.5 LP spaces 291
7.6 Further limit theorems for martingales 297
7.7 Exchangeability, De Finetti's theorem 306
References 312
8. TOPICS IN THE THEORY OF STOCHASTIC PROCESSES 315
8.1 Foundations and basic concepts 315
8.2 Poisson process 321
8.3 Renewal processes 328
8.4 The Galton Watson process; busy periods in queues 336
8.5 Markov chains 341
8.6 An invariance principle 353
References 359
Hints for Solutions of Exercises 361
Index 397 |
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| spelling | Galambos, János 1940-2019 Verfasser (DE-588)108371271 aut Advanced probability theory 1. print. New York [u.a.] Dekker 1988 VII, 405 S. txt rdacontent n rdamedia nc rdacarrier Probability, pure and applied 3 Probabilités Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s DE-604 Wahrscheinlichkeitstheorie (DE-588)4079013-7 s 2\p DE-604 Probability, pure and applied 3 (DE-604)BV000844355 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015341311&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Galambos, János 1940-2019 Advanced probability theory Probability, pure and applied Probabilités Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4064324-4 (DE-588)4151278-9 |
| title | Advanced probability theory |
| title_auth | Advanced probability theory |
| title_exact_search | Advanced probability theory |
| title_exact_search_txtP | Advanced probability theory |
| title_full | Advanced probability theory |
| title_fullStr | Advanced probability theory |
| title_full_unstemmed | Advanced probability theory |
| title_short | Advanced probability theory |
| title_sort | advanced probability theory |
| topic | Probabilités Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Probabilités Probabilities Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015341311&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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