Probability:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2007
|
| Series: | Graduate studies in mathematics
80 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 224 S. 27 cm |
| ISBN: | 0821842153 9780821842157 |
Staff View
MARC
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| 245 | 1 | 0 | |a Probability |c Davar Khoshnevisan |
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Record in the Search Index
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| adam_text | Contents
Preface
................................... xi
General Notation.............................. xv
Chapter
1.
Classical Probability
..................... 1
§1.
Discrete Probability
........................ 1
§2.
Conditional Probability
...................... 4
§3.
Independence
............................ 6
§4.
Discrete Distributions
....................... 6
§5.
Absolutely Continuous Distributions
............... 10
§6.
Expectation and Variance
..................... 12
Problems
................................. 13
Notes
................................... 15
Chapter
2.
Bernoulli Trials
....................... 17
§1.
The Classical Theorems
...................... 18
Problems
................................. 21
Notes
................................... 22
Chapter
3.
Measure Theory
....................... 23
§1.
Measure Spaces
.......................... 23
§2.
Lebesgue Measure
......................... 25
§3.
Completion
............................. 28
§4.
Proof of
Carathéodory s
Theorem
................ 30
Problems
................................. 33
vii
viii Contents
Notes
................................... 34
Chapter
4.
Integration
.......................... 35
§1.
Measurable Functions
....................... 35
§2.
The Abstract Integral
....................... 37
§3.
¿^-Spaces
.............................. 39
§4.
Modes of Convergence
....................... 43
§5.
Limit Theorems
.......................... 45
§6.
The
Radon-Nikodým
Theorem
.................. 47
Problems
................................. 49
Notes
................................... 52
Chapter
5.
Product Spaces
........................ 53
§1.
Finite Products
.......................... 53
§2.
Infinite Products
.......................... 58
§3.
Complement: Proof of Kolmogorov s Extension Theorem
... 60
Problems
................................. 62
Notes
................................... 64
Chapter
6.
Independence
......................... 65
§1.
Random Variables and Distributions
............... 65
§2.
Independent Random Variables
.................. 67
§3.
An Instructive Example
...................... 71
§4.
Khintchine s Weak Law of Large Numbers
........... 71
§5.
Kolmogorov s Strong Law of Large Numbers
.......... 73
§6.
Applications
............................ 77
Problems
................................. 84
Notes
................................... 89
Chapter
7.
The Central Limit Theorem
................ 91
§1.
Weak Convergence
......................... 91
§2.
Weak Convergence and Compact-Support Functions
...... 94
§3.
Harmonic Analysis in Dimension One
.............. 96
§4.
The Plancherel Theorem
..................... 97
§5.
The 1-D Central Limit Theorem
................. 100
§6.
Complements to the CLT
..................... 101
Problems
.................................
Ill
Notes
................................... 117
Contents
їх
Chapter
8.
Martingales
.......................... 119
§1.
Conditional Expectations
..................... 119
§2.
Filtrations and Semi-Martingales
................. 126
§3.
Stopping Times and Optional Stopping
............. 129
§4.
Applications to Random Walks
.................. 131
§5.
Inequalities and Convergence
................... 134
§6.
Further Applications
........................ 136
Problems
................................. 151
Notes
................................... 157
Chapter
9.
Brownian Motion
......................159
§1.
Gaussian Processes
........................160
§2.
Wiener s Construction: Brownian Motion on
[0,1).......165
§3.
Nowhere-Differentiability
.....................168
§4.
The Brownian Filtration and Stopping Times
..........170
§5.
The Strong Markov Property
...................173
§6.
The Reflection Principle
......................175
Problems
.................................176
Notes
...................................180
Chapter
10.
Terminus: Stochastic Integration
.............181
§1.
The Indefinite
Ito
Integral
....................181
§2.
Continuous Martingales in L2(P)
.................187
§3.
The Definite
Ito
Integral
.....................189
§4.
Quadratic Variation
........................192
§5.
Itô s
Formula and Two Applications
...............193
Problems
.................................199
Notes
...................................201
Appendix
..................................203
§1.
Hubert Spaces
...........................203
§2.
Fourier Series
............................205
Bibliography
................................209
Index
....................................217
|
| adam_txt |
Contents
Preface
. xi
General Notation. xv
Chapter
1.
Classical Probability
. 1
§1.
Discrete Probability
. 1
§2.
Conditional Probability
. 4
§3.
Independence
. 6
§4.
Discrete Distributions
. 6
§5.
Absolutely Continuous Distributions
. 10
§6.
Expectation and Variance
. 12
Problems
. 13
Notes
. 15
Chapter
2.
Bernoulli Trials
. 17
§1.
The Classical Theorems
. 18
Problems
. 21
Notes
. 22
Chapter
3.
Measure Theory
. 23
§1.
Measure Spaces
. 23
§2.
Lebesgue Measure
. 25
§3.
Completion
. 28
§4.
Proof of
Carathéodory's
Theorem
. 30
Problems
. 33
vii
viii Contents
Notes
. 34
Chapter
4.
Integration
. 35
§1.
Measurable Functions
. 35
§2.
The Abstract Integral
. 37
§3.
¿^-Spaces
. 39
§4.
Modes of Convergence
. 43
§5.
Limit Theorems
. 45
§6.
The
Radon-Nikodým
Theorem
. 47
Problems
. 49
Notes
. 52
Chapter
5.
Product Spaces
. 53
§1.
Finite Products
. 53
§2.
Infinite Products
. 58
§3.
Complement: Proof of Kolmogorov's Extension Theorem
. 60
Problems
. 62
Notes
. 64
Chapter
6.
Independence
. 65
§1.
Random Variables and Distributions
. 65
§2.
Independent Random Variables
. 67
§3.
An Instructive Example
. 71
§4.
Khintchine's Weak Law of Large Numbers
. 71
§5.
Kolmogorov's Strong Law of Large Numbers
. 73
§6.
Applications
. 77
Problems
. 84
Notes
. 89
Chapter
7.
The Central Limit Theorem
. 91
§1.
Weak Convergence
. 91
§2.
Weak Convergence and Compact-Support Functions
. 94
§3.
Harmonic Analysis in Dimension One
. 96
§4.
The Plancherel Theorem
. 97
§5.
The 1-D Central Limit Theorem
. 100
§6.
Complements to the CLT
. 101
Problems
.
Ill
Notes
. 117
Contents
їх
Chapter
8.
Martingales
. 119
§1.
Conditional Expectations
. 119
§2.
Filtrations and Semi-Martingales
. 126
§3.
Stopping Times and Optional Stopping
. 129
§4.
Applications to Random Walks
. 131
§5.
Inequalities and Convergence
. 134
§6.
Further Applications
. 136
Problems
. 151
Notes
. 157
Chapter
9.
Brownian Motion
.159
§1.
Gaussian Processes
.160
§2.
Wiener's Construction: Brownian Motion on
[0,1).165
§3.
Nowhere-Differentiability
.168
§4.
The Brownian Filtration and Stopping Times
.170
§5.
The Strong Markov Property
.173
§6.
The Reflection Principle
.175
Problems
.176
Notes
.180
Chapter
10.
Terminus: Stochastic Integration
.181
§1.
The Indefinite
Ito
Integral
.181
§2.
Continuous Martingales in L2(P)
.187
§3.
The Definite
Ito
Integral
.189
§4.
Quadratic Variation
.192
§5.
Itô's
Formula and Two Applications
.193
Problems
.199
Notes
.201
Appendix
.203
§1.
Hubert Spaces
.203
§2.
Fourier Series
.205
Bibliography
.209
Index
.217 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:39:14Z |
| indexdate | 2024-07-09T21:06:49Z |
| institution | BVB |
| isbn | 0821842153 9780821842157 |
| language | English |
| lccn | 2006052603 |
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| physical | XVI, 224 S. 27 cm |
| publishDate | 2007 |
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| publisher | American Math. Soc. |
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| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Khoshnevisan, Davar 1964- Verfasser (DE-588)124139795 aut Probability Davar Khoshnevisan Providence, RI American Math. Soc. 2007 XVI, 224 S. 27 cm txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 80 Probabilités Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Maßtheorie (DE-588)4074626-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Maßtheorie (DE-588)4074626-4 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1157-2 Graduate studies in mathematics 80 (DE-604)BV009739289 80 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016023961&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Khoshnevisan, Davar 1964- Probability Graduate studies in mathematics Probabilités Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4074626-4 (DE-588)4123623-3 |
| title | Probability |
| title_auth | Probability |
| title_exact_search | Probability |
| title_exact_search_txtP | Probability |
| title_full | Probability Davar Khoshnevisan |
| title_fullStr | Probability Davar Khoshnevisan |
| title_full_unstemmed | Probability Davar Khoshnevisan |
| title_short | Probability |
| title_sort | probability |
| topic | Probabilités Probabilities Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Probabilités Probabilities Wahrscheinlichkeitstheorie Maßtheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016023961&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT khoshnevisandavar probability |