Probability theory
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Courant Inst. of Math. Sciences [u.a.]
2001
|
| Schriftenreihe: | Courant lecture notes in mathematics
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
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| 245 | 1 | 0 | |a Probability theory |c S. R. S. Varadhan |
| 264 | 1 | |a New York, NY |b Courant Inst. of Math. Sciences [u.a.] |c 2001 | |
| 300 | |a VII, 167 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Courant lecture notes in mathematics |v 7 | |
| 650 | 4 | |a Wahrscheinlichkeitstheorie | |
| 650 | 0 | 7 | |a Wahrscheinlichkeitstheorie |0 (DE-588)4079013-7 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 001z 2001 B 941 0202 MAT 600f 2003 A 119 |
|---|---|
| DE-BY-TUM_katkey | 1393129 |
| DE-BY-TUM_location | 01 02 |
| DE-BY-TUM_media_number | 040071421697 040020712294 |
| _version_ | 1820809286984400896 |
| adam_text | Chapter
1.
Measure Theory
1.1.
Introduction
1.2.
Construction of Measures
1.3.
Integration
1.4.
Transformations
1.5.
Product Spaces
Contents
Preface
vii
1
1
3
7
13
14
1.6.
Distributions and Expectations
16
Chapter
2.
Weak Convergence
19
2.1.
Characteristic Functions
19
2.2.
Moment-Generating Functions
22
2.3.
Weak Convergence
24
Chapter
3.
Independent Sums
35
3.1.
Independence and Convolution
35
3.2.
Weak Law of Large Numbers
37
3.3.
Strong Limit Theorems
40
3.4.
Series of Independent Random Variables
43
3.5.
Strong Law of Large Numbers
48
3.6.
Central Limit Theorem
49
3.7.
Accompanying Laws
54
3.8.
Infinitely Divisible Distributions
59
3.9.
Laws of the Iterated Logarithm
66
Chapter
4.
Dependent Random Variables
73
4.1.
Conditioning
73
4.2.
Conditional Expectation
79
4.3.
Conditional Probability
81
4.4.
Markov Chains
84
4.5.
Stopping Times and Renewal Times
89
4.6.
Countable State Space
90
4.7.
Some Examples
98
Chapter
5.
Martingales
109
5.1.
Definitions and Properties
109
5.2.
Martingale Convergence Theorems
112
vi
CONTENTS
5.3.
Doob Decomposition Theorem
115
5.4.
Stopping Times
117
5.5.
Up-crossing Inequality
120
5.6.
Martingale Transforms, Option Pricing
121
5.7.
Martingales and Markov Chains
123
Chapter
6.
Stationary Stochastic Processes
131
6.1.
Ergodic Theorems
131
6.2.
Structure of Stationary Measures
135
6.3.
Stationary Markov Processes
137
6.4.
Mixing Properties of Markov Processes
141
6.5.
Central Limit Theorem for Martingales
143
6.6.
Stationary Gaussian Processes
147
Chapter
7.
Dynamic Programming and Filtering
157
7.1.
Optimal Control
157
7.2.
Optimal Stopping
158
7.3.
Filtering
161
Bibliography
163
Index
165
This volume presents topics in probability theory covered during a
first-year graduate course given at the
Courant
Institute of
Mathematical Sciences. The necessary background material in
measure theory is developed, including the standard topics, such as
extension theorem, construction of measures, integration, product
spaces, Radon-Nikodym theorem, and conditional expectation.
In the first part of the book, characteristic functions are introduced,
followed by the study of weak convergence of probability distribu¬
tions. Then both the weak and strong limit theorems for sums of
independent random variables are proved, including the weak and
strong laws of large numbers, central limit theorems, laws of the
iterated logarithm, and the Kolmogorov three series theorem. The
first part concludes with infinitely divisible distributions and limit
theorems for sums of uniformly infinitesimal independent random
variables.
The second part of the book mainly deals with dependent random
variables, particularly martingales and Markov chains. Topics
include standard results regarding discrete parameter martingales
and Doob s inequalities. The standard topics in Markov chains are
treated, i.e., transience, and null and positive recurrence. A varied
collection of examples is given to demonstrate the connection
between martingales and Markov chains.
Additional topics covered in the book include stationary Gaussian
processes, ergodic theorems, dynamic programming, optimal stop¬
ping, and filtering. A large number of examples and exercises is
included. The book is a suitable text for a first-year graduate course
in probability.
|
| any_adam_object | 1 |
| author | Varadhan, S. R. S. 1940- |
| author_GND | (DE-588)13051957X |
| author_facet | Varadhan, S. R. S. 1940- |
| author_role | aut |
| author_sort | Varadhan, S. R. S. 1940- |
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| bvnumber | BV014002202 |
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| classification_rvk | SK 800 |
| classification_tum | MAT 600f |
| ctrlnum | (OCoLC)248030490 (DE-599)BVBBV014002202 |
| 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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| id | DE-604.BV014002202 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T15:45:43Z |
| institution | BVB |
| isbn | 0821828525 9780821828526 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009585839 |
| oclc_num | 248030490 |
| open_access_boolean | |
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| physical | VII, 167 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Courant Inst. of Math. Sciences [u.a.] |
| record_format | marc |
| series | Courant lecture notes in mathematics |
| series2 | Courant lecture notes in mathematics |
| spellingShingle | Varadhan, S. R. S. 1940- Probability theory Courant lecture notes in mathematics Wahrscheinlichkeitstheorie Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4079013-7 |
| title | Probability theory |
| title_auth | Probability theory |
| title_exact_search | Probability theory |
| title_full | Probability theory S. R. S. Varadhan |
| title_fullStr | Probability theory S. R. S. Varadhan |
| title_full_unstemmed | Probability theory S. R. S. Varadhan |
| title_short | Probability theory |
| title_sort | probability theory |
| topic | Wahrscheinlichkeitstheorie Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009585839&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009585839&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012714106 |
| work_keys_str_mv | AT varadhansrs probabilitytheory |