| 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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