Probability theory: a comprehensive course
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English German |
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London
Springer
[2020]
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| Ausgabe: | Third edition |
| Schriftenreihe: | Universitext
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xiv, 716 Seiten Illustrationen |
| ISBN: | 9783030564018 |
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| adam_text | Contents 1 Basic Measure Theory........................................................................... 1.1 Classes of Sets............................................................................ 1.2 Set Functions............................................................................... 1.3 The Measure Extension Theorem................................................ 1.4 Measurable Maps......................................................................... 1.5 Random Variables........................................................................ 1 1 11 18 36 45 2 Independence......................................................................................... 2.1 Independence of Events.............................................................. 2.2 Independent Random Variables.................................................. 2.3 Kolmogorov’s 0-1 Law............................................................... 2.4 Example: Percolation.................................................................. 53 53 61 69 73 3 Generating Functions............................................................................ 3.1 Definition and Examples............................................................ 3.2 Poisson Approximation............................................................... 3.3 Branching Processes................................................................... 85 85 89 91 4 The Integral........................ 95 4.1 Construction and Simple Properties........................................... 95 4.2 Monotone Convergence and Fatou’s
Lemma.............................. 104 4.3 Lebesgue Integral Versus RiemannIntegral................................. 107 5 Moments and Laws of Large Numbers............................................... 5.1 Moments ..................................................................................... 5.2 Weak Law of Large Numbers..................................................... 5.3 Strong Law of Large Numbers.................................................... 5.4 Speed of Convergence in the Strong LLN.................................. 5.5 The Poisson Process................................................................... 113 113 121 125 135 139 xi
xii Contents 6 Convergence Theorems......................................................................... 6.1 Almost Sure and Measure Convergence..................................... 6.2 Uniform Integrability.................................................................. 6.3 Exchanging Integral and Differentiation..................................... 147 147 153 160 7 Lp-Spaces and the Radon-Nikodym Theorem.................................... 7.1 Definitions................................................................................... 7.2 Inequalities and the Fischer-Riesz Theorem.............................. 7.3 Hilbert Spaces.............................................................................. 7.4 Lebesgue’s Decomposition Theorem.......................................... 7.5 Supplement: Signed Measures.................................................... 7.6 Supplement: Dual Spaces............................................................ 163 163 165 172 175 179 186 8 Conditional Expectations...................................................................... 8.1 Elementary Conditional Probabilities.......................................... 8.2 Conditional Expectations............................................................ 8.3 Regular Conditional Distribution................................................ 191 191 195 203 9 Martingales............................................................................................. 9.1 Processes, Filtrations, Stopping Times........................................ 9.2
Martingales.................................................................................. 9.3 Discrete Stochastic Integral........................................................ 9.4 Discrete Martingale Representation Theorem and the CRR Model........................................................................................... 213 213 218 223 10 Optional Sampling Theorems............................................................... 10.1 Doob Decomposition and Square Variation................................ 10.2 Optional Sampling and Optional Stopping................................. 10.3 Uniform Integrability and Optional Sampling............................. 229 229 233 239 11 Martingale Convergence Theorems and Their Applications............. 11.1 Doob’s Inequality........................................................................ 11.2 Martingale Convergence Theorems............................................. 11.3 Example: Branching Process....................................................... 241 241 243 254 12 Backwards Martingales and Exchangeability..................................... 12.1 Exchangeable Families of Random Variables............................. 12.2 Backwards Martingales............................................................... 12.3 De Finetti’s Theorem.................................................................. 257 257 263 266 13 273 274 281 290 300 Convergence of Measures....................................................................... 13.1 A Topology
Primer..................................................................... 13.2 Weak and Vague Convergence.................................................... 13.3 Prohorov’s Theorem.................................................................... 13.4 Application: A Fresh Look at de Finetti’s Theorem.................... 224
Contents xiii 14 Probability Measures on Product Spaces............................................ 14.1 Product Spaces............................................................................ 14.2 Finite Products and Transition Kernels....................................... 14.3 Kolmogorov’s Extension Theorem.............................................. 14.4 Markov Semigroups.................................................................... 303 304 307 317 322 15 Characteristic Functions and the Central Limit Theorem................. 15.1 Separating Classes of Functions.................................................. 15.2 Characteristic Functions: Examples............................................ 15.3 Levy’s Continuity Theorem........................................................ 15.4 Characteristic Functions and Moments....................................... 15.5 The Central Limit Theorem........................................................ 15.6 Multidimensional Central Limit Theorem.................................. 327 327 336 344 349 356 365 16 Infinitely Divisible Distributions........................................................... 367 16.1 Lévy-Khinchin Formula.............................................................. 367 16.2 Stable Distributions..................................................................... 381 17 Markov Chains...................................................................................... 17.1 Definitions and Construction...................................................... 17.2 Discrete Markov Chains:
Examples............................................ 17.3 Discrete Markov Processes in Continuous Time........................ 17.4 Discrete Markov Chains: Recurrence and Transience................ 17.5 Application: Recurrence and Transience of Random Walks....... 17.6 Invariant Distributions................................................................. 17.7 Stochastic Ordering and Coupling................................ 391 391 399 404 411 415 423 429 18 Convergence of Markov Chains............................................................ 18.1 Periodicity of Markov Chains..................................... 18.2 Coupling and Convergence Theorem...................... 18.3 Markov Chain Monte Carlo Method...................... 18.4 Speed of Convergence................................................................. 435 435 439 445 453 19 Markov Chains and Electrical Networks............................................ 19.1 Harmonic Functions................................................................... 19.2 Reversible Markov Chains......................................................... 19.3 Finite Electrical Networks........................................................... 19.4 Recurrence and Transience......................................................... 19.5 Network Reduction..................................................................... 19.6 Random Walk in a Random Environment.................................. 461 462 465 467 473 480 488 20 Ergodic Theory...................................................................................... 20.1
Definitions................................................................................... 20.2 Ergodic Theorems...................................................................... 20.3 Examples..................................................................................... 20.4 Application: Recurrence of Random Walks................................ 20.5 Mixing......................................................................................... 20.6 Entropy........................................................................................ 493 493 497 500 502 506 510
XIV Contents 21 Brownian Motion......................................................................................... 21.1 Continuous Versions........................................................................ 21.2 Construction and Path Properties................................................... 21.3 Strong Markov Property.................................................................. 21.4 Supplement: Feller Processes......................................................... 21.5 Construction via ^-Approximation.............................................. 21.6 The Space C([0, oo))...................................................................... 21.7 Convergence of Probability Measures onC([0, oo)) .................... 21.8 Donsker’s Theorem.......................................................................... 21.9 Pathwise Convergence of Branching Processes............................ 21.10 Square Variation and Local Martingales........................................ 515 515 522 529 532 535 544 546 549 553 560 22 Law of the Iterated Logarithm.................................................................. 22.1 Iterated Logarithm for the Brownian Motion.................................. 22.2 Skorohod’s Embedding Theorem................................................... 22.3 Hartman-Wintner Theorem............................................................ 573 573 576 583 23 Large Deviations.......................................................................................... 23.1 Cramer’s
Theorem........................................................................... 23.2 Large Deviations Principle............................................................. 23.3 Sanov’s Theorem.............................................................................. 23.4 Varadhan’s Lemma and Free Energy.............................................. 587 588 594 598 603 24 The Poisson Point Process.......................................................................... 24.1 Random Measures........................................................................... 24.2 Properties of the Poisson Point Process......................................... 24.3 The Poisson-Dirichlet Distribution................................................ 611 611 616 627 25 The Ito Integral............................................................................................ 635 25.1 Itô Integral with Respect to Brownian Motion............................. 635 25.2 Itô Integral with Respect to Diffusions......................................... 644 25.3 The Itô Formula............................................................................... 648 25.4 Dirichlet Problem and Brownian Motion...................................... 657 25.5 Recurrence and Transience of BrownianMotion........................... 659 26 Stochastic Differential Equations............................................................. 26.1 Strong Solutions.............................................................................. 26.2 Weak Solutions and the Martingale
Problem................................ 26.3 Weak Uniqueness via Duality......................................................... 665 665 675 682 References.............................................................................................................. 691 Notation Index...................................................................................................... 699 Name Index........................................................................................................... 703 Subject Index........................................................................................................ 707
|
| adam_txt |
Contents 1 Basic Measure Theory. 1.1 Classes of Sets. 1.2 Set Functions. 1.3 The Measure Extension Theorem. 1.4 Measurable Maps. 1.5 Random Variables. 1 1 11 18 36 45 2 Independence. 2.1 Independence of Events. 2.2 Independent Random Variables. 2.3 Kolmogorov’s 0-1 Law. 2.4 Example: Percolation. 53 53 61 69 73 3 Generating Functions. 3.1 Definition and Examples. 3.2 Poisson Approximation. 3.3 Branching Processes. 85 85 89 91 4 The Integral. 95 4.1 Construction and Simple Properties. 95 4.2 Monotone Convergence and Fatou’s
Lemma. 104 4.3 Lebesgue Integral Versus RiemannIntegral. 107 5 Moments and Laws of Large Numbers. 5.1 Moments . 5.2 Weak Law of Large Numbers. 5.3 Strong Law of Large Numbers. 5.4 Speed of Convergence in the Strong LLN. 5.5 The Poisson Process. 113 113 121 125 135 139 xi
xii Contents 6 Convergence Theorems. 6.1 Almost Sure and Measure Convergence. 6.2 Uniform Integrability. 6.3 Exchanging Integral and Differentiation. 147 147 153 160 7 Lp-Spaces and the Radon-Nikodym Theorem. 7.1 Definitions. 7.2 Inequalities and the Fischer-Riesz Theorem. 7.3 Hilbert Spaces. 7.4 Lebesgue’s Decomposition Theorem. 7.5 Supplement: Signed Measures. 7.6 Supplement: Dual Spaces. 163 163 165 172 175 179 186 8 Conditional Expectations. 8.1 Elementary Conditional Probabilities. 8.2 Conditional Expectations. 8.3 Regular Conditional Distribution. 191 191 195 203 9 Martingales. 9.1 Processes, Filtrations, Stopping Times. 9.2
Martingales. 9.3 Discrete Stochastic Integral. 9.4 Discrete Martingale Representation Theorem and the CRR Model. 213 213 218 223 10 Optional Sampling Theorems. 10.1 Doob Decomposition and Square Variation. 10.2 Optional Sampling and Optional Stopping. 10.3 Uniform Integrability and Optional Sampling. 229 229 233 239 11 Martingale Convergence Theorems and Their Applications. 11.1 Doob’s Inequality. 11.2 Martingale Convergence Theorems. 11.3 Example: Branching Process. 241 241 243 254 12 Backwards Martingales and Exchangeability. 12.1 Exchangeable Families of Random Variables. 12.2 Backwards Martingales. 12.3 De Finetti’s Theorem. 257 257 263 266 13 273 274 281 290 300 Convergence of Measures. 13.1 A Topology
Primer. 13.2 Weak and Vague Convergence. 13.3 Prohorov’s Theorem. 13.4 Application: A Fresh Look at de Finetti’s Theorem. 224
Contents xiii 14 Probability Measures on Product Spaces. 14.1 Product Spaces. 14.2 Finite Products and Transition Kernels. 14.3 Kolmogorov’s Extension Theorem. 14.4 Markov Semigroups. 303 304 307 317 322 15 Characteristic Functions and the Central Limit Theorem. 15.1 Separating Classes of Functions. 15.2 Characteristic Functions: Examples. 15.3 Levy’s Continuity Theorem. 15.4 Characteristic Functions and Moments. 15.5 The Central Limit Theorem. 15.6 Multidimensional Central Limit Theorem. 327 327 336 344 349 356 365 16 Infinitely Divisible Distributions. 367 16.1 Lévy-Khinchin Formula. 367 16.2 Stable Distributions. 381 17 Markov Chains. 17.1 Definitions and Construction. 17.2 Discrete Markov Chains:
Examples. 17.3 Discrete Markov Processes in Continuous Time. 17.4 Discrete Markov Chains: Recurrence and Transience. 17.5 Application: Recurrence and Transience of Random Walks. 17.6 Invariant Distributions. 17.7 Stochastic Ordering and Coupling. 391 391 399 404 411 415 423 429 18 Convergence of Markov Chains. 18.1 Periodicity of Markov Chains. 18.2 Coupling and Convergence Theorem. 18.3 Markov Chain Monte Carlo Method. 18.4 Speed of Convergence. 435 435 439 445 453 19 Markov Chains and Electrical Networks. 19.1 Harmonic Functions. 19.2 Reversible Markov Chains. 19.3 Finite Electrical Networks. 19.4 Recurrence and Transience. 19.5 Network Reduction. 19.6 Random Walk in a Random Environment. 461 462 465 467 473 480 488 20 Ergodic Theory. 20.1
Definitions. 20.2 Ergodic Theorems. 20.3 Examples. 20.4 Application: Recurrence of Random Walks. 20.5 Mixing. 20.6 Entropy. 493 493 497 500 502 506 510
XIV Contents 21 Brownian Motion. 21.1 Continuous Versions. 21.2 Construction and Path Properties. 21.3 Strong Markov Property. 21.4 Supplement: Feller Processes. 21.5 Construction via ^-Approximation. 21.6 The Space C([0, oo)). 21.7 Convergence of Probability Measures onC([0, oo)) . 21.8 Donsker’s Theorem. 21.9 Pathwise Convergence of Branching Processes. 21.10 Square Variation and Local Martingales. 515 515 522 529 532 535 544 546 549 553 560 22 Law of the Iterated Logarithm. 22.1 Iterated Logarithm for the Brownian Motion. 22.2 Skorohod’s Embedding Theorem. 22.3 Hartman-Wintner Theorem. 573 573 576 583 23 Large Deviations. 23.1 Cramer’s
Theorem. 23.2 Large Deviations Principle. 23.3 Sanov’s Theorem. 23.4 Varadhan’s Lemma and Free Energy. 587 588 594 598 603 24 The Poisson Point Process. 24.1 Random Measures. 24.2 Properties of the Poisson Point Process. 24.3 The Poisson-Dirichlet Distribution. 611 611 616 627 25 The Ito Integral. 635 25.1 Itô Integral with Respect to Brownian Motion. 635 25.2 Itô Integral with Respect to Diffusions. 644 25.3 The Itô Formula. 648 25.4 Dirichlet Problem and Brownian Motion. 657 25.5 Recurrence and Transience of BrownianMotion. 659 26 Stochastic Differential Equations. 26.1 Strong Solutions. 26.2 Weak Solutions and the Martingale
Problem. 26.3 Weak Uniqueness via Duality. 665 665 675 682 References. 691 Notation Index. 699 Name Index. 703 Subject Index. 707 |
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| spelling | Klenke, Achim Verfasser (DE-588)1036340058 aut Wahrscheinlichkeitstheorie Probability theory a comprehensive course Achim Klenke Third edition London Springer [2020] © 2020 xiv, 716 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Universitext Cálculo de probabilidade larpcal Probabilidade larpcal Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-030-56402-5 (DE-604)BV046974483 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032411587&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Klenke, Achim Probability theory a comprehensive course Cálculo de probabilidade larpcal Probabilidade larpcal Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4123623-3 |
| title | Probability theory a comprehensive course |
| title_alt | Wahrscheinlichkeitstheorie |
| title_auth | Probability theory a comprehensive course |
| title_exact_search | Probability theory a comprehensive course |
| title_exact_search_txtP | Probability theory a comprehensive course |
| title_full | Probability theory a comprehensive course Achim Klenke |
| title_fullStr | Probability theory a comprehensive course Achim Klenke |
| title_full_unstemmed | Probability theory a comprehensive course Achim Klenke |
| title_short | Probability theory |
| title_sort | probability theory a comprehensive course |
| title_sub | a comprehensive course |
| topic | Cálculo de probabilidade larpcal Probabilidade larpcal Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Cálculo de probabilidade Probabilidade Probabilities Wahrscheinlichkeitstheorie Lehrbuch |
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