A probability path:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
2014
|
| Series: | Modern Birkhäuser classics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Reprint der Birkhäuser Edition von 2005 Literaturverz. S. 443 - 444 |
| Physical Description: | xiv, 453 Seiten Diagramme |
| ISBN: | 9780817684082 |
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Record in the Search Index
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| adam_text | Titel: A probability path
Autor: Resnick, Sidney I
Jahr: 2014
Contents
Preface xii
1 Sets and Events 1
1.1 Introduction 1
1.2 Basic Set Theory 2
1.2.1 Indicator functions..........................................5
1.3.........limits of Sets............. ......, . . ...............6
1.4 Monotone Sequences................................................8
1.5 Set Operations and Closure..........................................11
1.5.1 Examples....................................................13
1.6 The cr-field Generated by a Given Class C..........................15
1.7 Borel Sets on the Real Line..........................................16
1.8 Comparing Borel Sets................................................18
1.9 Exercises ............................................................20
2. Probability Spaces 29
2.1 Basic Definitions and Properties....................................29
2.2 More on Closure ..........................35
2.2.1 Dynkin s theorem......................36
2.2.2 Proof of Dynkin s theorem.................38
2.3 Two Constructions..................................................40
2.4 Constructions of Probability Spaces................................42
2.4.1 General Construction of a Probability Model.......43
2.4.2 Proof of the Second Extension Theorem..........49
2.5 Measure Constructions.......................57
2.5.1 Lebesgue Measure on (0,1]................57
2.5.2 Construction of a Probability Measure on R with Given
Distribution Function F(x).................61
2.6 Exercises..............................63
3 Random Variables, Elements, and Measurable Maps 71
3.1 Inverse Maps............................71
3.2 Measurable Maps, Random Elements,
Induced Probability Measures ...................74
3.2.1 Composition........................77
3.2.2 Random Elements of Metric Spaces............78
3.2.3 Measurability and Continuity ...............80
3.2.4 Measurability and Limits..................81
3.3 a-Fields Generated by Maps....................83
3.4 Exercises..............................85
4 Independence 91
4.1 Basic Definitions..........................91
4.2 Independent Random Variables...................93
4.3 1 vo Examples of Independence..................95
4.3.1 Records, Ranks, Renyi Theorem..............95
4.3.2 Dyadic Expansions of Uniform Random Numbers .... 98
4.4 More on Independence: Groupings.................100
4.5 Independence, Zero-One Laws, Borel-Cantelli Lemma......102
4.5.1 Borel-Cantelli Lemma...................102
4.5.2 Bore! Zero-One Law....................103
4.5.3 Kolmogorov Zero-One Law................107
4.6 Exercises..............................110
5 Integration and Expectation 117
5.1 Preparation for Integration.....................117
5.1.1 Simple Functions......................117
5.1.2 Measurability and Simple Functions............118
5.2 Expectation and Integration.....................119
5.2.1 Expectation of Simple Functions..............119
5.2.2 Extension of the Definition.................122
5.2.3 Basic Properties of Expectation ..............123
5.3 Limits and Integrals.........................131
5.4 Indefinite Integrals.........................134
5.5 The Transformation Theorem and Densities............135
5.5.1 Expectation is Always an Integral on R..........137
5.5.2 Densities..........................139
5.6 The Riemann vs Lebesgue Integral.................139
5.7 Product Spaces...........................143
5.8 Probability Measures on Product Spaces..............147
5.9 Fubini s theorem ..........................149
5.10 Exercises..............................155
6 Convergence Concepts 167
6.1 Almost Sure Convergence .....................167
6.2 Convergence in Probability.....................169
6.2.1 Statistical Terminology...................170
6.3 Connections Between a.s. and i.p. Convergence..........171
6.4 Quantile Estimation.........................178
6.5 Lp Convergence...........................180
6.5.1 Uniform Integrability....................182
6.5.2 Interlude: A Review of Inequalities............186
6.6 More on Lp Convergence......................189
6.7 Exercises ..............................195
7 Laws of Large Numbers and Sums
of Independent Random Variables 203
7.1 Truncation and Equivalence.....................203
7.2 A General Weak Law of Large Numbers..............204
7.3 Almost Sure Convergence of Sums
of Independent Random Variables.................209
7.4 Strong Laws of Large Numbers...................213
7.4.1 Two Examples.......................215
7.5 The Strong Law of Large Numbers for IID Sequences.......219
7.5.1 Two Applications of the SLLN...............222
7.6 The Kolmogorov Three Series Theorem..............226
7.6.1 Necessity of the Kolmogorov Three Series Theorem . . .230
7.7 Exercises..............................234
8 Convergence in Distribution 247
8.1 Basic Definitions..........................247
8.2 Scheffé s lemma...........................252
8.2.1 Scheffé s lemma and Order Statistics ...........255
8.3 The Baby Skorohod Theorem....................258
8.3.1 The Delta Method .....................261
8.4 Weak Convergence Equivalences; Portmanteau Theorem.....263
8.5 More Relations Among Modes of Convergence..........267
8.6 New Convergences from Old....................268
8.6.1 Example: The Central Limit Theorem for m-Dependent
Random Variables.....................270
8.7 The Convergence to Types Theorem................274
8.7.1 Application of Convergence to Types: Limit Distributions
for Extremes........................278
8.8 Exercises ..............................282
9 Characteristic Functions and the Central Limit Theorem 293
9.1 Review of Moment Generating Functions
and the Central Limit Theorem...................294
9.2 Characteristic Functions: Definition and First Properties......295
9.3 Expansions.............................297
9.3.1 Expansion of éx......................297
9.4 Moments and Derivatives......................301
9.5 Two Big Theorems: Uniqueness and Continuity..........302
9.6 The Selection Theorem, Tightness, and
Prohorov s theorem.........................307
9.6.1 The Selection Theorem...................307
9.6.2 Tightness, Relative Compactness,
and Prohorov s theorem..................309
9.6.3 Proof of the Continuity Theorem..............311
9.7 The Classical CLT for lid Random Variables............312
9.8 The Lindeberg-Feller CLT.....................314
9.9 Exercises..............................321
10 Martingales 333
10.1 Prelude to Conditional Expectation:
The Radon-Nikodym Theorem..................333
10.2 Definition of Conditional Expectation...............339
10.3 Properties of Conditional Expectation...............344
10.4 Martingales ............................353
10.5 Examples of Martingales .....................356
10.6 Connections between Martingales and Submartingales......360
10.6.1 Doob s Decomposition .................360
10.7 Stopping Times..........................363
10.8 Positive Super Martingales ....................366
10.8.1 Operations on Supermartingales.............367
10.8.2 Upcrossings.......................369
10.8.3 Boundedness Properties.................369
10.8.4 Convergence of Positive Super Martingales.......371
10.8.5 Closure..........................374
10.8.6 Stopping Supermartingales...............377
10.9 Examples..............................379
10.9.1 Gambler s Ruin .....................379
10.9.2 Branching Processes...................380
10.9.3 Some Differentiation Theory ..............382
10.10 Martingale and Submartingale Convergence...........386
10.10.1 Krickeberg Decomposition...............386
10.10.2 Doob s (Sub)martingale Convergence Theorem.....387
10.11 Regularity and Closure ......................388
10.12 Regularity and Stopping......................390
10.13 Stopping Theorems........................392
10.14 Wald s Identity and Random Walks................398
10.14.1 The Basic Martingales..................400
10.14.2 Regular Stopping Times.................402
10.14.3 Examples of Integrable Stopping Times.........407
10.14.4 The Simple Random Walk................409
10.15 Reversed Martingales.......................412
10.16 Fundamental Theorems of Mathematical Finance ........416
10.16.1 A Simple Market Model.................416
10.16.2 Admissible Strategies and Arbitrage ..........419
10.16.3 Arbitrage and Martingales................420
10.16.4 Complete Markets....................425
10.16.5 Option Pricing......................428
10.17 Exercises..............................429
References 443
Index 445
|
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| author | Resnick, Sidney I. 1945- |
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Resnick, Sidney I. 1945- Verfasser (DE-588)111457114 aut A probability path Sidney I. Resnick New York Springer 2014 xiv, 453 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Modern Birkhäuser classics Reprint der Birkhäuser Edition von 2005 Literaturverz. S. 443 - 444 Wahrscheinlichkeitstheorie Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Erscheint auch als Online-Ausgabe 978-0-8176-8409-9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026916657&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Resnick, Sidney I. 1945- A probability path Wahrscheinlichkeitstheorie Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4079013-7 |
| title | A probability path |
| title_auth | A probability path |
| title_exact_search | A probability path |
| title_full | A probability path Sidney I. Resnick |
| title_fullStr | A probability path Sidney I. Resnick |
| title_full_unstemmed | A probability path Sidney I. Resnick |
| title_short | A probability path |
| title_sort | a probability path |
| topic | Wahrscheinlichkeitstheorie Probabilities Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Probabilities Wahrscheinlichkeitsrechnung |
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