Preface | p. xi |
Abbreviations and Notations | p. xiii |
A Review of Probability Distributions and Their Properties | p. 1 |
Introduction | p. 1 |
The Exponential Density | p. 1 |
The Gamma Density | p. 2 |
The Beta Density | p. 3 |
The Uniform Density | p. 5 |
The Cauchy Density | p. 5 |
The Normal Density in One Dimension | p. 6 |
Convolution Property | p. 6 |
The Normal Density in n Dimensions | p. 7 |
Infinitely Divisible Distributions | p. 9 |
Stable Distributions | p. 11 |
Problems for Solution | p. 12 |
Definition and Characteristics of a Stochastic Process | p. 19 |
Introduction | p. 19 |
Analytic Definition | p. 19 |
Definition in Terms of Finite-Dimensional Distributions | p. 20 |
Moments of Stochastic Processes | p. 23 |
Some Problems in Stochastic Processes | p. 24 |
Probability Models | p. 25 |
Comments on the Definition of a Stochastic Process | p. 26 |
Some Important Classes of Stochastic Processes | p. 29 |
Stationary Processes | p. 29 |
Processes with Stationary Independent Increments | p. 31 |
Markov Processes | p. 33 |
Problems for Solution | p. 37 |
Stationary Processes | p. 41 |
Examples of Real Stationary Processes | p. 41 |
The General Case | p. 44 |
A Second Order Calculus for Stationary Processes | p. 46 |
Time Series Models | p. 54 |
Mean Square Convergence | p. 57 |
Problems for Solution | p. 58 |
The Brownian Motion and the Poisson Process: Levy Processes | p. 63 |
The Brownian Motion | p. 63 |
Historical Remarks | p. 63 |
Introduction | p. 63 |
Properties of the Brownian Motion | p. 65 |
The Poisson Process | p. 71 |
Introduction | p. 71 |
Properties of the Poisson Process | p. 72 |
The Compound Poisson Process | p. 78 |
Levy Processes | p. 80 |
The Gaussian Process | p. 81 |
Application to Brownian Storage Models | p. 89 |
The Inverse Gaussian Process | p. 91 |
The Randomized Bernoulli Random Walk | p. 91 |
Application to the Simple Queue | p. 96 |
Levy Processes: Further Properties | p. 97 |
Problems for Solution | p. 103 |
Renewal Processes and Random Walks | p. 107 |
Renewal Processes: Introduction | p. 107 |
Physical Interpretation | p. 108 |
The Renewal-Counting Processes {N(t)} | p. 110 |
Renewal Theorems | p. 122 |
The Age and the Remaining Lifetime | p. 125 |
The Stationary Renewal Process | p. 130 |
The Case of the Infinite Mean | p. 131 |
The Random Walk on the Real Line: Introduction | p. 134 |
The Maximum and Minimum Functionate | p. 135 |
Ladder Processes | p. 139 |
Limit Theorems for M[subscript n] | p. 147 |
Problems for Solution | p. 150 |
Martingales in Discrete Time | p. 155 |
Introduction and Examples | p. 155 |
Some Terminology | p. 158 |
Martingales Relative to a Sigma-Field | p. 159 |
Decision Functions; Optional Stopping | p. 161 |
Submartingales and Supermartingales | p. 162 |
Optional Skipping and Sampling Theorems | p. 168 |
Application to Random Walks | p. 177 |
Convergence Properties | p. 181 |
The Concept of Fairness | p. 185 |
Problems for Solution | p. 186 |
Branching Processes | p. 189 |
Introduction | p. 189 |
The Problem of Extinction | p. 194 |
The Extinction Time and the Total Progeny | p. 197 |
The Supercritical Case | p. 200 |
Estimation | p. 203 |
Problems for Solution | p. 207 |
Regenerative Phenomena | p. 213 |
Introduction | p. 213 |
Discrete Time Regenerative Phenomena | p. 216 |
Subordination of Renewal Counting Processes | p. 221 |
The Simple Random Walk in D Dimensions | p. 225 |
The Bernoulli Random Walk | p. 227 |
Ladder Sets of Random Walks on the Real Line | p. 232 |
Further Examples of Recurrent Phenomena | p. 237 |
Regenerative Phenomena in Continuous Time | p. 241 |
Stable Regenerative Phenomena | p. 255 |
Problems for Solution | p. 258 |
Markov Chains | p. 261 |
Introduction | p. 261 |
Discrete Time Markov Chains | p. 261 |
Examples of Finite Markov Chains | p. 263 |
Markov Trials | p. 263 |
The Bernoulli-Laplace Diffusion Model | p. 267 |
The Limit Distribution of Finite Markov Chains | p. 268 |
Classification of States. Limit Theorems | p. 272 |
Closed Sets. Irreducible Chains | p. 275 |
Stationary Distributions | p. 279 |
Examples of Infinite Markov Chains | p. 281 |
The Branching Process as a Markov Chain | p. 281 |
The Queueing System GI/M/1 | p. 283 |
Continuous Time Markov Chains | p. 286 |
Examples of Continuous Time Markov Chains | p. 292 |
The Poisson Process as a Markov Chain | p. 292 |
The Pure Birth Process | p. 293 |
The Pure Death Process | p. 296 |
The Birth and Death Process | p. 297 |
Models for Population Growth | p. 298 |
Some Deterministic Models | p. 298 |
Stochastic Models | p. 301 |
The Yule-Furry Model | p. 302 |
The Feller-Arley Model | p. 304 |
The Kendall Model | p. 308 |
The Differential Equation | p. 309 |
Problems for Solution | p. 310 |
Finite Markov Chains | p. 310 |
Infinite Markov Chains | p. 314 |
Continuous Time Markov Chains | p. 315 |
Models for Population Growth | p. 317 |
Tauberian Theorems | p. 321 |
Some Asymptotic Relations | p. 337 |
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