Basic stochastic processes: the Mark Kac lectures
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Macmillan [u.a.]
1988
|
| Edition: | 1. print. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 258 S. |
| ISBN: | 0023598204 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
CHAPTER ZERO
Preliminaries
0.1 Random Variables 4
0.2 The Moment Generating Function 7
0.3 Discrete Random Variables 9
0.4 Continuous Random Variables 12
0.5 Families of Random Variables, Independence 15
0.6 Multivariate Normal Distribution 21
0.7 Sums of Independent Random Variables 26
0.8 Convergence of Random Variables 31
Problems 43
CHAPTER ONE
The Fourier Transform
and the Fourier Inversion Formula
1.1 An Integral Representation of the SGN Function 49
1.2 Fourier Inversion of Unit Rectangle Functions 52
1.3 Step Functions and Beyond, Application to Density Functions 54
1.4 A Digression About the Dirac 8 function 55
1.5 An Example of a Fourier Inversion 58
1.6 The Multidimensional Fourier Transform and Inversion Formula 60
xi
Xll Contents
1.7 Characteristic Functions: Applications of the Fourier Inversion
Formula to Probability Theory 60
1.8 Multidimensional Characteristic Function and Its Inversion Formula 69
Problems 73
CHAPTER TWO
Fourier Series
2.1 The Mean Square Approximation 78
2.2 Fourier Series 80
2.3 Convergence of Fourier Series, Consequences 82
2.4 Functions of Arbitrary Period 85
2.5 Fourier Series for Arbitrary Functions 86
2.6 The Complex Form of the Fourier Series 87
2.7 The Fourier Inversion Formula 88
2.8 Plancherel s Formula 89
2.9 The Geometry of Fourier Series, Hilbert Spaces 91
2.10 Nonnegative Definite Functions 95
Problems 96
CHAPTER THREE
Poisson Processes
3.1 Introduction to the Poisson Process 103
3.2 An Equivalent Derivation 105
3.3 An Alternative Definition of the Poisson Process 106
3.4 Poisson Process from Exponential Random Variables 107
3.5 Poisson Process from Uniform Random Variables 110
3.6 Poisson Process and the Order Statistics 113
3.7 Poisson Process and the Binomial Distribution 114
3.8 Generalizations of the Poisson Process 116
Problems 125
CHAPTER FOUR
Shot Noise
4.1 Shot Noise and Its Statistical Moments 129
4.2 A Related Limiting Distribution 135
4.3 The Limiting Distribution of the Shot Noise Process 136
4.4 Random Noise 139
4.5 The Spectral Density of Random Noise 141
Problems 142
Contents X1U
CHAPTER FIVE
Gaussian Processes
5.1 The Gaussian Process 148
5.2 Calculation of the Characteristic Function 150
5.3 The Covariance Matrix 151
5.4 A Rule for Calculating Higher Correlations 152
5.5 The Integral of a Gaussian Process 154
5.6 The Wiener Process 157
5.7 Stationary Processes, Spectral Density 159
Problems 165
CHAPTER SIX
Markov Gaussian Processes
6.1 Markov Processes 170
6.2 Doob s Theorem 171
6.3 The Chapman Kolmogorov Equation 174
6.4 Completion of Doob s Theorem 175
6.5 The Ornstein Uhlenbeck Process 176
6.6 Conditional Expectation and Doob s Theorem 177
Problems 180
CHAPTER SEVEN
Brownian Motion
7.1 Einstein s Approach 183
7.2 The Velocity of the Brownian Particle 184
7.3 The Displacement of the Brownian Particle 185
7.4 Brownian Motion 187
7.5 The Fokker Planck Equation 189
7.6 The Fokker Planck Equation and Brownian Motion 191
7.7 Stochastic Differential Equations 194
Problems 198
CHAPTER EIGHT
Markov Chains
8.1 The Ehrenfest Dog Flea Model 201
8.2 Markov Chains 203
8.3 Properties of Stochastic Matrices 205
8.4 Applications of Diagonalization 208
XIV Contents
8.5 Two Other Methods for Finding Limiting Distributions 212
8.6 Random Walk 214
8.7 Random Walk with Absorbing Barriers 216
8.8 The Stationary Distribution and Mean Recurrence Time 222
8.9 The Ehrenfest Model Revisited, Elastic Random Walk 227
8.10 Branching Processes 228
8.11 Characterizing Transient and Recurrent States 232
8.12 Ergodicity 234
8.13 Convergence to the Equilibrium Distribution 236
Problems 238
APPENDIX
Review of Linear Algebra
A.I Matrices and Matrix Algebra 244
A.2 Systems of Linear Equations, Linear Transformations 247
A.3 Square Matrices 248
A.4 Structure of Square Matrices 250
A.5 Symmetric Matrices 253
A.6 Quadratic Forms 253
Index
255
|
| adam_txt |
Contents
CHAPTER ZERO
Preliminaries
0.1 Random Variables 4
0.2 The Moment Generating Function 7
0.3 Discrete Random Variables 9
0.4 Continuous Random Variables 12
0.5 Families of Random Variables, Independence 15
0.6 Multivariate Normal Distribution 21
0.7 Sums of Independent Random Variables 26
0.8 Convergence of Random Variables 31
Problems 43
CHAPTER ONE
The Fourier Transform
and the Fourier Inversion Formula
1.1 An Integral Representation of the SGN Function 49
1.2 Fourier Inversion of Unit Rectangle Functions 52
1.3 Step Functions and Beyond, Application to Density Functions 54
1.4 A Digression About the Dirac 8 function 55
1.5 An Example of a Fourier Inversion 58
1.6 The Multidimensional Fourier Transform and Inversion Formula 60
xi
Xll Contents
1.7 Characteristic Functions: Applications of the Fourier Inversion
Formula to Probability Theory 60
1.8 Multidimensional Characteristic Function and Its Inversion Formula 69
Problems 73
CHAPTER TWO
Fourier Series
2.1 The Mean Square Approximation 78
2.2 Fourier Series 80
2.3 Convergence of Fourier Series, Consequences 82
2.4 Functions of Arbitrary Period 85
2.5 Fourier Series for Arbitrary Functions 86
2.6 The Complex Form of the Fourier Series 87
2.7 The Fourier Inversion Formula 88
2.8 Plancherel's Formula 89
2.9 The Geometry of Fourier Series, Hilbert Spaces 91
2.10 Nonnegative Definite Functions 95
Problems 96
CHAPTER THREE
Poisson Processes
3.1 Introduction to the Poisson Process 103
3.2 An Equivalent Derivation 105
3.3 An Alternative Definition of the Poisson Process 106
3.4 Poisson Process from Exponential Random Variables 107
3.5 Poisson Process from Uniform Random Variables 110
3.6 Poisson Process and the Order Statistics 113
3.7 Poisson Process and the Binomial Distribution 114
3.8 Generalizations of the Poisson Process 116
Problems 125
CHAPTER FOUR
Shot Noise
4.1 Shot Noise and Its Statistical Moments 129
4.2 A Related Limiting Distribution 135
4.3 The Limiting Distribution of the Shot Noise Process 136
4.4 Random Noise 139
4.5 The Spectral Density of Random Noise 141
Problems 142
Contents X1U
CHAPTER FIVE
Gaussian Processes
5.1 The Gaussian Process 148
5.2 Calculation of the Characteristic Function 150
5.3 The Covariance Matrix 151
5.4 A Rule for Calculating Higher Correlations 152
5.5 The Integral of a Gaussian Process 154
5.6 The Wiener Process 157
5.7 Stationary Processes, Spectral Density 159
Problems 165
CHAPTER SIX
Markov Gaussian Processes
6.1 Markov Processes 170
6.2 Doob's Theorem 171
6.3 The Chapman Kolmogorov Equation 174
6.4 Completion of Doob's Theorem 175
6.5 The Ornstein Uhlenbeck Process 176
6.6 Conditional Expectation and Doob's Theorem 177
Problems 180
CHAPTER SEVEN
Brownian Motion
7.1 Einstein's Approach 183
7.2 The Velocity of the Brownian Particle 184
7.3 The Displacement of the Brownian Particle 185
7.4 Brownian Motion 187
7.5 The Fokker Planck Equation 189
7.6 The Fokker Planck Equation and Brownian Motion 191
7.7 Stochastic Differential Equations 194
Problems 198
CHAPTER EIGHT
Markov Chains
8.1 The Ehrenfest Dog Flea Model 201
8.2 Markov Chains 203
8.3 Properties of Stochastic Matrices 205
8.4 Applications of Diagonalization 208
XIV Contents
8.5 Two Other Methods for Finding Limiting Distributions 212
8.6 Random Walk 214
8.7 Random Walk with Absorbing Barriers 216
8.8 The Stationary Distribution and Mean Recurrence Time 222
8.9 The Ehrenfest Model Revisited, Elastic Random Walk 227
8.10 Branching Processes 228
8.11 Characterizing Transient and Recurrent States 232
8.12 Ergodicity 234
8.13 Convergence to the Equilibrium Distribution 236
Problems 238
APPENDIX
Review of Linear Algebra
A.I Matrices and Matrix Algebra 244
A.2 Systems of Linear Equations, Linear Transformations 247
A.3 Square Matrices 248
A.4 Structure of Square Matrices 250
A.5 Symmetric Matrices 253
A.6 Quadratic Forms 253
Index
255 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:04:04Z |
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| isbn | 0023598204 |
| language | English |
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| spelling | Iranpour, Reza Verfasser aut Basic stochastic processes the Mark Kac lectures 1. print. New York Macmillan [u.a.] 1988 XIV, 258 S. txt rdacontent n rdamedia nc rdacarrier Processus stochastiques ram Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s DE-604 Chacon, Paul Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015103756&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Iranpour, Reza Chacon, Paul Basic stochastic processes the Mark Kac lectures Processus stochastiques ram Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4057630-9 |
| title | Basic stochastic processes the Mark Kac lectures |
| title_auth | Basic stochastic processes the Mark Kac lectures |
| title_exact_search | Basic stochastic processes the Mark Kac lectures |
| title_exact_search_txtP | Basic stochastic processes the Mark Kac lectures |
| title_full | Basic stochastic processes the Mark Kac lectures |
| title_fullStr | Basic stochastic processes the Mark Kac lectures |
| title_full_unstemmed | Basic stochastic processes the Mark Kac lectures |
| title_short | Basic stochastic processes |
| title_sort | basic stochastic processes the mark kac lectures |
| title_sub | the Mark Kac lectures |
| topic | Processus stochastiques ram Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Processus stochastiques Stochastic processes Stochastischer Prozess |
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