Stochastic processes: basic theory and its applications
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2007
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 341 S. graph. Darst. |
| ISBN: | 9789812706263 9812706267 |
Staff View
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| 245 | 1 | 0 | |a Stochastic processes |b basic theory and its applications |c Narahari U. Prabhu |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2007 | |
| 300 | |a XIV, 341 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
| _version_ | 1804136561093115904 |
|---|---|
| adam_text | Contents
Preface
xi
Abbreviations and Notations
xiii
1.
A Review of Probability Distributions and Their Properties
1
1.1
Introduction
.......................... 1
1.2
The Exponential Density
................... 1
1.3
The Gamma Density
..................... 2
1.4
The Beta Density
....................... 3
1.5
The Uniform Density
..................... 5
1.6
The Cauchy Density
..................... 5
1.7
The Normal Density in One Dimension
...........
G
1.7.1
Convolution Property
................. 6
1.8
The Normal Density in
η
Dimensions
............ 7
1.9
Infinitely Divisible Distributions
............... 9
1.10
Stable Distributions
...................... 11
1.11
Problems for Solution
..................... 12
2.
Definition and Characteristics of a Stochastic Process
19
2.1
Introduction
.......................... 19
2.2
Analytic Definition
...................... 19
2.3
Definition in Terms of Finite-Dimensional Distributions
. . 20
2.4
Moments of Stochastic Processes
............... 23
2.5
Some Problems in Stochastic Processes
........... 24
2.6
Probability Models
...................... 25
2.7
Comments on the Definition of a Stochastic Process
.... 26
29
3.
Some Important Classes of Stochastic Processes
3.1
Stationary Processes
..................... 29
3.2
Processes with Stationary Independent Increments
.... 31
3.3
Markov Processes
....................... 33
.................. 37
3.4
Problems for Solution
4.
Stationary Processes
4.1
4.2
4.3
4.4
4.5
4.6
5.
The
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Examples of Real Stationary Processes
......
The General Case
..................
A Second Order Calculus for Stationary Processes
Time Series Models
.................
Mean Square Convergence
.............
Problems for Solution
................
41
41
44
46
54
57
58
63
63
63
63
65
71
71
72
78
80
81
89
91
91
96
97
103
107
Renewal Processes: Introduction
............... 107
6.1.1
Physical Interpretation
................ 108
The Renewal-Counting Processes {N(t)}
.......... 110
Renewal Theorems
...................... 122
The Age and the Remaining Lifetime
............ 125
Brownian Motion and the
Poisson
Process: Levy Processes
The Brownian Motion
....................
5.1.1
Historical Remarks
..................
5.1.2
Introduction
......................
5.1.3
Properties of the Brownian Motion
.........
The
Poisson
Process
.....................
5.2.1
Introduction
......................
5.2.2
Properties of the
Poisson
Process
..........
5.2.3
The Compound
Poisson
Process
...........
Levy Processes
.........................
The Gaussian Process
.....................
5.4.1
Application to Brownian Storage Models
......
The Inverse Gaussian Process
................
The Randomized Bernoulli Random Walk
.........
5.6.1
Application to the Simple Queue
...........
Levy Processes: Further Properties
.............
Problems for Solution
.....................
6.
Renewal Processes and Random Walks
6.1
6.2
6.3
6.4
6.5
The Stationary Renewal Process
............... 130
6.6
The Case of the Infinite Mean
...... .......... 131
6.7
The Random Walk on the Real Line: Introduction
..... 134
6.8
The Maximum and Minimum Functional
......... 135
6.9
Ladder Processes
....................... 139
6.10
Limit Theorems for Mn
.................... 147
6.11
Problems for Solution
..................... 150
7.
Martingales in Discrete Time
155
7.1
Introduction and Examples
.................. 155
7.2
Some Terminology
....................... 158
7.3
Martingales Relative to
a
Sigma-Field
............ 159
7.4
Decision Functions; Optional Stopping
........... 161
7.5
Submartingales
and
Supermartingales
............ 162
7.6
Optional
Skipping and Sampling Theorems
......... 168
7.7
Application to Random Walks
................ 177
7.8
Convergence Properties
.................... 181
7.9
The Concept of Fairness
................... 185
7.10
Problems for Solution
..................... 186
8.
Branching Processes
189
8.1
Introduction
.......................... 189
8.2
The Problem of Extinction
.................. 194
8.3
The Extinction Time and the Total Progeny
........ 197
8.4
The Supercritical Case
.................... 200
8.5
Estimation
........................... 203
8.6
Problems for Solution
..................... 207
9.
Regenerative Phenomena
213
9.1
Introduction
.......................... 213
9.2
Discrete Time Regenerative Phenomena
.......... 216
9.3
Subordination of Renewal Counting Processes
....... 221
9.4
The Simple Random Walk in
D
Dimensions
........ 225
9.5
The Bernoulli Random Walk
................. 227
9.6
Ladder Sets of Random Walks on the Real Line
...... 232
9.7
Further Examples of Recurrent Phenomena
........ 237
9.8
Regenerative Phenomena in Continuous Time
....... 241
9.9
Stable Regenerative Phenomena
............... 255
9.10
Problems for Solution
..................... 258
|
| adam_txt |
Contents
Preface
xi
Abbreviations and Notations
xiii
1.
A Review of Probability Distributions and Their Properties
1
1.1
Introduction
. 1
1.2
The Exponential Density
. 1
1.3
The Gamma Density
. 2
1.4
The Beta Density
. 3
1.5
The Uniform Density
. 5
1.6
The Cauchy Density
. 5
1.7
The Normal Density in One Dimension
.
G
1.7.1
Convolution Property
. 6
1.8
The Normal Density in
η
Dimensions
. 7
1.9
Infinitely Divisible Distributions
. 9
1.10
Stable Distributions
. 11
1.11
Problems for Solution
. 12
2.
Definition and Characteristics of a Stochastic Process
19
2.1
Introduction
. 19
2.2
Analytic Definition
. 19
2.3
Definition in Terms of Finite-Dimensional Distributions
. . 20
2.4
Moments of Stochastic Processes
. 23
2.5
Some Problems in Stochastic Processes
. 24
2.6
Probability Models
. 25
2.7
Comments on the Definition of a Stochastic Process
. 26
29
3.
Some Important Classes of Stochastic Processes
3.1
Stationary Processes
. 29
3.2
Processes with Stationary Independent Increments
. 31
3.3
Markov Processes
. 33
. 37
3.4
Problems for Solution
4.
Stationary Processes
4.1
4.2
4.3
4.4
4.5
4.6
5.
The
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Examples of Real Stationary Processes
.
The General Case
.
A Second Order Calculus for Stationary Processes
Time Series Models
.
Mean Square Convergence
.
Problems for Solution
.
41
41
44
46
54
57
58
63
63
63
63
65
71
71
72
78
80
81
89
91
91
96
97
103
107
Renewal Processes: Introduction
. 107
6.1.1
Physical Interpretation
. 108
The Renewal-Counting Processes {N(t)}
. 110
Renewal Theorems
. 122
The Age and the Remaining Lifetime
. 125
Brownian Motion and the
Poisson
Process: Levy Processes
The Brownian Motion
.
5.1.1
Historical Remarks
.
5.1.2
Introduction
.
5.1.3
Properties of the Brownian Motion
.
The
Poisson
Process
.
5.2.1
Introduction
.
5.2.2
Properties of the
Poisson
Process
.
5.2.3
The Compound
Poisson
Process
.
Levy Processes
.
The Gaussian Process
.
5.4.1
Application to Brownian Storage Models
.
The Inverse Gaussian Process
.
The Randomized Bernoulli Random Walk
.
5.6.1
Application to the Simple Queue
.
Levy Processes: Further Properties
.
Problems for Solution
.
6.
Renewal Processes and Random Walks
6.1
6.2
6.3
6.4
6.5
The Stationary Renewal Process
. 130
6.6
The Case of the Infinite Mean
.". 131
6.7
The Random Walk on the Real Line: Introduction
. 134
6.8
The Maximum and Minimum Functional
. 135
6.9
Ladder Processes
. 139
6.10
Limit Theorems for Mn
. 147
6.11
Problems for Solution
. 150
7.
Martingales in Discrete Time
155
7.1
Introduction and Examples
. 155
7.2
Some Terminology
. 158
7.3
Martingales Relative to
a
Sigma-Field
. 159
7.4
Decision Functions; Optional Stopping
. 161
7.5
Submartingales
and
Supermartingales
. 162
7.6
Optional
Skipping and Sampling Theorems
. 168
7.7
Application to Random Walks
. 177
7.8
Convergence Properties
. 181
7.9
The Concept of Fairness
. 185
7.10
Problems for Solution
. 186
8.
Branching Processes
189
8.1
Introduction
. 189
8.2
The Problem of Extinction
. 194
8.3
The Extinction Time and the Total Progeny
. 197
8.4
The Supercritical Case
. 200
8.5
Estimation
. 203
8.6
Problems for Solution
. 207
9.
Regenerative Phenomena
213
9.1
Introduction
. 213
9.2
Discrete Time Regenerative Phenomena
. 216
9.3
Subordination of Renewal Counting Processes
. 221
9.4
The Simple Random Walk in
D
Dimensions
. 225
9.5
The Bernoulli Random Walk
. 227
9.6
Ladder Sets of Random Walks on the Real Line
. 232
9.7
Further Examples of Recurrent Phenomena
. 237
9.8
Regenerative Phenomena in Continuous Time
. 241
9.9
Stable Regenerative Phenomena
. 255
9.10
Problems for Solution
. 258 |
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| author | Prabhu, Narahari U. 1924- |
| author_GND | (DE-588)118076183 |
| author_facet | Prabhu, Narahari U. 1924- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2/3 |
| dewey-search | 519.2/3 |
| dewey-sort | 3519.2 13 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV022473055 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:45:23Z |
| indexdate | 2024-07-09T20:58:21Z |
| institution | BVB |
| isbn | 9789812706263 9812706267 |
| language | English |
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| physical | XIV, 341 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Prabhu, Narahari U. 1924- Verfasser (DE-588)118076183 aut Stochastic processes basic theory and its applications Narahari U. Prabhu Singapore [u.a.] World Scientific 2007 XIV, 341 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Stochastic processes Textbooks Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015680509&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Prabhu, Narahari U. 1924- Stochastic processes basic theory and its applications Stochastic processes Textbooks Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4057630-9 |
| title | Stochastic processes basic theory and its applications |
| title_auth | Stochastic processes basic theory and its applications |
| title_exact_search | Stochastic processes basic theory and its applications |
| title_exact_search_txtP | Stochastic processes basic theory and its applications |
| title_full | Stochastic processes basic theory and its applications Narahari U. Prabhu |
| title_fullStr | Stochastic processes basic theory and its applications Narahari U. Prabhu |
| title_full_unstemmed | Stochastic processes basic theory and its applications Narahari U. Prabhu |
| title_short | Stochastic processes |
| title_sort | stochastic processes basic theory and its applications |
| title_sub | basic theory and its applications |
| topic | Stochastic processes Textbooks Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Stochastic processes Textbooks Stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015680509&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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