Measure theory, probability, and stochastic processes:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2022]
|
| Series: | Graduate texts in mathematics
295 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 406 Seiten Diagramme |
| ISBN: | 9783031142048 9783031142079 |
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| 650 | 4 | |a Probability Theory | |
| 650 | 4 | |a Stochastic Processes | |
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Record in the Search Index
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| adam_text | Contents Part I 1 Measurable Spaces...................................................................................... 1.1 Measurable Sets ........................................................................... 1.2 1.3 1.4 1.5 2 Integration of Nonnegative Functions ........................................... Integrable Functions ...................................................................... Integrals Depending on a Parameter .............................................. Exercises ........................................................................................ Construction of Measures .......................................................................... 3.1 Outer Measures .............................................................................. 3.2 3.3 3.4 3.5 3.6 3.7 4 Positive Measures........................................................................... Measurable Functions ................................................................... Monotone Class............................................................................. Exercises ........................................................................................ Integration of Measurable Functions........................................................ 2.1 2.2 2.3 2.4 3 Measure Theory Lebesgue Measure ......................................................................... Relation with Riemann Integrals.................................................... A Subset of R Which Is Not Measurable....................................... Finite Measures on R and the
Stieltjes Integral ............................. The Riesz-Markov-Kakutani Representation Theorem ................. Exercises ....................................................................................... LP Spaces....................................................................................................... 4.1 4.2 4.3 4.4 4.5 Definitions and the Hôlder Inequality ........................................... The Banach Space LP(E, A, μ) .................................................... Density Theorems in Lp Spaces .................................................... The Radon-Nikodym Theorem ..................................................... Exercises ....................................................................................... 3 3 6 10 13 15 17 17 27 31 35 41 41 44 53 55 56 58 59 63 63 67 71 75 81
5 Product Measures ................................................................................... 85 5.1 Product σ-Fields ........................................................................... 85 5.2 Product Measures .......................................................................... 87 5.3 The Fubini Theorems.................................................................... 90 5.4 Applications.................................................................................. 94 5.4.1 Integration by Parts ........................................................ 94 5.4.2 Convolution .................................................................... 95 5.4.3 The Volume of the Unit Ball .......................................... 99 5.5 Exercises ....................................................................................... 101 6 Signed Measures...................................................................................... 6.1 Definition and Total Variation....................................................... 6.2 The Jordan Decomposition........................................................... 6.3 The Duality Between Lp and Lq .................................................. 6.4 The Riesz-Markov- Kakutani Representation Theorem for Signed Measures........................................................................... 6.5 Exercises ...................................................................................... 118 119 Change of Variables................................................................................ 7.1 The
Change of Variables Formula ................................................ 7.2 The Gamma Function .................................................................. 7.3 Lebesgue Measure on the Unit Sphere ........................................ 7.4 Exercises ...................................................................................... 121 121 127 128 130 7 105 105 109 113 Part II Probability Theory 8 Foundations of Probability Theory....................................................... 8.1 General Definitions ....................................................................... 8.1.1 Probability Spaces .......................................................... 8.1.2 Random Variables............................................................ 8.1.3 Mathematical Expectation ............................................... 8.1.4 An Example: Bertrand’s Paradox ................................... 8.1.5 Classical Laws ................................................................ 8.1.6 Distribution Function of a Real Random Variable ......... 8.1.7 The σ-Field Generated by a Random Variable ............... 8.2 Moments of Random Variables ..................................................... 8.2.1 Moments and Variance ................................................... 8.2.2 Linear Regression ............................................................ 8.2.3 Characteristic Functions .................................................. 8.2.4 Laplace Transform and Generating Functions................ 8.3 Exercises
........................................................................................ 135 136 136 138 140 144 146 149 150 151 151 155 156 160 162 9 Independence ......................................................................................... 9.1 Independent Events ........................................................................ 9.2 Independence for σ-Fields and Random Variables ...................... 9.3 The Borel-Cantelli Lemma............................................................ 9.4 Construction of Independent Sequences........................................ 167 168 169 177 181
9.5 9.6 9.7 9.8 10 Convergence in Distribution ....................................................... Two Applications ......................................................................... 10.4.1 The Convergence of Empirical Measures....................... 10.4.2 The Central Limit Theorem ........................................... 10.4.3 The Multidimensional Central Limit Theorem .............. Exercises ..................................................................................... 209 216 216 219 221 223 Conditioning ................................................................................................ 227 227 230 230 233 237 238 242 242 242 244 248 251 10.5 11.1 11.2 11.3 11.4 11.5 11.6 Part III 12 Discrete Conditioning ................................................................. The Definition of Conditional Expectation .................................. 11.2.1 Integrable Random Variables .......................................... 11.2.2 Nonnegative Random Variables ..................................... 11.2.3 The Special Case of Square Integrable Variables .......... Specific Properties of the Conditional Expectation ..................... Evaluation of Conditional Expectation ........................................ 11.4.1 Discrete Conditioning .................................................... 11.4.2 Random Variables with a Density ................................... 11.4.3 Gaussian Conditioning .................................................. Transition Probabilities and ConditionalDistributions ................ Exercises
...................................................................................... Stochastic Processes Theory of Martingales................................................................................ 257 12.1 Definitions and Examples ........................................................... 12.2 Stopping Times ........................................................................... 12.3 Almost Sure Convergence of Martingales ..................................... 12.4 Convergence in Lp When p 1 .................................................. 12.5 Uniform Integrability and Martingales ......................................... 12.6 Optional Stopping Theorems ..................................................... 12.7 Backward Martingales................................................................. 12.8 Exercises ..................................................................................... 13 182 186 188 195 Convergence of Random Variables.......................................................... 199 10.1 The Different Notions of Convergence ........................................ 199 10.2 The Strong Law of Large Numbers ............................................. 204 10.3 10.4 11 Sums of Independent Random Variables ..................................... Convolution Semigroups ............................................................. The Poisson Process .................................................................... Exercises ...................................................................................... 257 263 266 274 280 284
290 296 Markov Chains ............................................................................................ 303 13.1 13.2 Definitions and First Properties ................................................... A Few Examples.......................................................................... 13.2.1 Independent Random Variables ..................................... 13.2.2 Random Walks on Zá ..................................................... 303 308 308 309
13.3 13.4 13.5 13.6 13.7 13.8 14 13.2.3 Simple Random Walk on a Graph.................... ............. 13.2.4 Galton-Watson Branching Processes.............................. The Canonical Markov Chain ...................................................... The Classification of States ......................................................... Invariant Measures....................................................................... Ergodic Theorems ....................................................................... Martingales and Markov Chains ................................................. Exercises ..................................................................................... 309 310 311 317 326 332 338 343 Brownian Motion......................................................................................... 349 Brownian Motion as a Limit of Random Walks.......................... The Construction of Brownian Motion ....................................... The Wiener Measure ................................................................... First Properties of Brownian Motion .......................................... The Strong Markov Property....................................................... Harmonic Functions and the Dirichlet Problem........................... Harmonic Functions and Brownian Motion ................................ Exercises ...................................................................................... 349 353 359 361 365 372 383 389 A Few Facts from Functional Analysis.................................................... 395
Notes and Suggestions for Further Reading .................................................. 399 References.............................................................................................................. 401 Index....................................................................................................................... 403 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 A
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| adam_txt |
Contents Part I 1 Measurable Spaces. 1.1 Measurable Sets . 1.2 1.3 1.4 1.5 2 Integration of Nonnegative Functions . Integrable Functions . Integrals Depending on a Parameter . Exercises . Construction of Measures . 3.1 Outer Measures . 3.2 3.3 3.4 3.5 3.6 3.7 4 Positive Measures. Measurable Functions . Monotone Class. Exercises . Integration of Measurable Functions. 2.1 2.2 2.3 2.4 3 Measure Theory Lebesgue Measure . Relation with Riemann Integrals. A Subset of R Which Is Not Measurable. Finite Measures on R and the
Stieltjes Integral . The Riesz-Markov-Kakutani Representation Theorem . Exercises . LP Spaces. 4.1 4.2 4.3 4.4 4.5 Definitions and the Hôlder Inequality . The Banach Space LP(E, A, μ) . Density Theorems in Lp Spaces . The Radon-Nikodym Theorem . Exercises . 3 3 6 10 13 15 17 17 27 31 35 41 41 44 53 55 56 58 59 63 63 67 71 75 81
5 Product Measures . 85 5.1 Product σ-Fields . 85 5.2 Product Measures . 87 5.3 The Fubini Theorems. 90 5.4 Applications. 94 5.4.1 Integration by Parts . 94 5.4.2 Convolution . 95 5.4.3 The Volume of the Unit Ball . 99 5.5 Exercises . 101 6 Signed Measures. 6.1 Definition and Total Variation. 6.2 The Jordan Decomposition. 6.3 The Duality Between Lp and Lq . 6.4 The Riesz-Markov- Kakutani Representation Theorem for Signed Measures. 6.5 Exercises . 118 119 Change of Variables. 7.1 The
Change of Variables Formula . 7.2 The Gamma Function . 7.3 Lebesgue Measure on the Unit Sphere . 7.4 Exercises . 121 121 127 128 130 7 105 105 109 113 Part II Probability Theory 8 Foundations of Probability Theory. 8.1 General Definitions . 8.1.1 Probability Spaces . 8.1.2 Random Variables. 8.1.3 Mathematical Expectation . 8.1.4 An Example: Bertrand’s Paradox . 8.1.5 Classical Laws . 8.1.6 Distribution Function of a Real Random Variable . 8.1.7 The σ-Field Generated by a Random Variable . 8.2 Moments of Random Variables . 8.2.1 Moments and Variance . 8.2.2 Linear Regression . 8.2.3 Characteristic Functions . 8.2.4 Laplace Transform and Generating Functions. 8.3 Exercises
. 135 136 136 138 140 144 146 149 150 151 151 155 156 160 162 9 Independence . 9.1 Independent Events . 9.2 Independence for σ-Fields and Random Variables . 9.3 The Borel-Cantelli Lemma. 9.4 Construction of Independent Sequences. 167 168 169 177 181
9.5 9.6 9.7 9.8 10 Convergence in Distribution . Two Applications . 10.4.1 The Convergence of Empirical Measures. 10.4.2 The Central Limit Theorem . 10.4.3 The Multidimensional Central Limit Theorem . Exercises . 209 216 216 219 221 223 Conditioning . 227 227 230 230 233 237 238 242 242 242 244 248 251 10.5 11.1 11.2 11.3 11.4 11.5 11.6 Part III 12 Discrete Conditioning . The Definition of Conditional Expectation . 11.2.1 Integrable Random Variables . 11.2.2 Nonnegative Random Variables . 11.2.3 The Special Case of Square Integrable Variables . Specific Properties of the Conditional Expectation . Evaluation of Conditional Expectation . 11.4.1 Discrete Conditioning . 11.4.2 Random Variables with a Density . 11.4.3 Gaussian Conditioning . Transition Probabilities and ConditionalDistributions . Exercises
. Stochastic Processes Theory of Martingales. 257 12.1 Definitions and Examples . 12.2 Stopping Times . 12.3 Almost Sure Convergence of Martingales . 12.4 Convergence in Lp When p 1 . 12.5 Uniform Integrability and Martingales . 12.6 Optional Stopping Theorems . 12.7 Backward Martingales. 12.8 Exercises . 13 182 186 188 195 Convergence of Random Variables. 199 10.1 The Different Notions of Convergence . 199 10.2 The Strong Law of Large Numbers . 204 10.3 10.4 11 Sums of Independent Random Variables . Convolution Semigroups . The Poisson Process . Exercises . 257 263 266 274 280 284
290 296 Markov Chains . 303 13.1 13.2 Definitions and First Properties . A Few Examples. 13.2.1 Independent Random Variables . 13.2.2 Random Walks on Zá . 303 308 308 309
13.3 13.4 13.5 13.6 13.7 13.8 14 13.2.3 Simple Random Walk on a Graph. . 13.2.4 Galton-Watson Branching Processes. The Canonical Markov Chain . The Classification of States . Invariant Measures. Ergodic Theorems . Martingales and Markov Chains . Exercises . 309 310 311 317 326 332 338 343 Brownian Motion. 349 Brownian Motion as a Limit of Random Walks. The Construction of Brownian Motion . The Wiener Measure . First Properties of Brownian Motion . The Strong Markov Property. Harmonic Functions and the Dirichlet Problem. Harmonic Functions and Brownian Motion . Exercises . 349 353 359 361 365 372 383 389 A Few Facts from Functional Analysis. 395
Notes and Suggestions for Further Reading . 399 References. 401 Index. 403 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 A |
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| index_date | 2024-07-03T20:57:06Z |
| indexdate | 2024-07-10T09:41:12Z |
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| spelling | Le Gall, Jean-François 1959- Verfasser (DE-588)1033682829 aut Measure theory, probability, and stochastic processes Jean-François Le Gall Cham Springer [2022] xiv, 406 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 295 Measure and Integration Probability Theory Stochastic Processes Measure theory Probabilities Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf Maßtheorie (DE-588)4074626-4 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Stochastischer Prozess (DE-588)4057630-9 s DE-604 Erscheint auch als Online-Ausgabe, eBook 978-3-031-14205-5 (DE-604)BV048541482 Graduate texts in mathematics 295 (DE-604)BV000000067 295 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=033926660&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Le Gall, Jean-François 1959- Measure theory, probability, and stochastic processes Graduate texts in mathematics Measure and Integration Probability Theory Stochastic Processes Measure theory Probabilities Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4057630-9 (DE-588)4079013-7 (DE-588)4074626-4 |
| title | Measure theory, probability, and stochastic processes |
| title_auth | Measure theory, probability, and stochastic processes |
| title_exact_search | Measure theory, probability, and stochastic processes |
| title_exact_search_txtP | Measure theory, probability, and stochastic processes |
| title_full | Measure theory, probability, and stochastic processes Jean-François Le Gall |
| title_fullStr | Measure theory, probability, and stochastic processes Jean-François Le Gall |
| title_full_unstemmed | Measure theory, probability, and stochastic processes Jean-François Le Gall |
| title_short | Measure theory, probability, and stochastic processes |
| title_sort | measure theory probability and stochastic processes |
| topic | Measure and Integration Probability Theory Stochastic Processes Measure theory Probabilities Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Measure and Integration Probability Theory Stochastic Processes Measure theory Probabilities Stochastic processes Stochastischer Prozess Wahrscheinlichkeitstheorie Maßtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033926660&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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