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Basic Notions | |
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Sample Space and Events | |
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Probabilities | |
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Counting Techniques | |
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Independence and Conditional Probability | |
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Independence | |
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Conditioning | |
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The Borel-Cantelli Theorem | |
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Discrete Random Variables | |
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Random Variables and Vectors | |
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Expected Value | |
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Variance and Other Moments | |
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Inequalities for Deviations | |
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Some Basic Distributions | |
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Convergence of Random Variables | |
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The Law of Large Numbers | |
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Conditional Expectation | |
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Generating Functions | |
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Branching Processes | |
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Random Walk Revisited | |
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Branching Processes | |
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Generating Functions | |
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Branching Processes Revisited | |
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More on Random Walk | |
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Markov Chains | |
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Definitions and Examples | |
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Probability Distributions of Markov Chains | |
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The First Step Analysis | |
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Passage Times | |
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Variables Defined on a Markov Chain | |
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Ergodicity and Stationary Distributions | |
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A Classification of States and Ergodicity | |
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Continuous Random Variables | |
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Continuous Distributions | |
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Some Basic Distributions | |
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Continuous Multivariate Distributions | |
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Sums of Independent Random Variables | |
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Conditional Distributions and Expectations | |
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Distributions in the General Case | |
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Simulation | |
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Distribution Functions | |
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Expected Values | |
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On Convergence in Distribution and Probability | |
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Simulation | |
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Histograms | |
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Moment Generating Functions | |
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Definitions and Properties | |
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Some Examples of Applications | |
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Exponential or Bernstein-Chernoff's Bounds | |
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The Central Limit Theorem for Independent Random Variables | |
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The Central Limit Theorem (CLT) for Independent and Identically Distributed Random Variables | |
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The CLT for Independent Variables in the General Case | |
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Covariance Analysis | |
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The Multivariate Normal Distribution | |
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The Multivariate Central Limit Theorem | |
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Covariance and Correlation | |
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Covariance Matrices and Some Applications | |
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The Multivariate Normal Distribution | |
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Maxima and Minima of Random Variables | |
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Elements of Reliability Theory | |
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Hazard Rate and Survival Probabilities | |
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Maxima and Minima of Random Variables | |
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Reliability Characteristics | |
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Limit Theorems for Maxima and Minima | |
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Hazard Rate | |
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Survival Probabilities | |
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Stochastic Processes: Preliminaries | |
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A General Definition | |
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Processes with Independent Increments | |
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Brownian Motion | |
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Markov Processes | |
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A Representation and Simulation of Markov Processes in Discrete Time | |
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Counting and Queuing Processes | |
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Birth and Death Processes: A General Scheme | |
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Poisson Processes | |
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Birth and Death Processes | |
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Elements of Renewal Theory | |
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Preliminaries | |
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Limit Theorems | |
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Some Proofs | |
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Martingales in Discrete Time | |
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Definitions and Properties | |
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Optional Time and Some Applications | |
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Martingales and a Financial Market Model | |
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Limit Theorems for Martingales | |
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Brownian Motion and Martingales in Continuous Time | |
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Brownian Motion and Its Generalizations | |
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Martingales in Continuous Time | |
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More on Dependency Structures | |
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Arrangement Structures and the Corresponding Dependencies | |
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Measures of Dependency | |
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Limit Theorems for Dependent Random Variables | |
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Symmetric Distributions | |
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De Finetti's Theorem | |
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Comparison of Random Variables | |
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Risk Evaluation | |
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Some Particular Criteria | |
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Expected Utility | |
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Generalizations of the EUM Criterion | |
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Appendix | |
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References | |
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Answers to Exercises | |
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Index | |
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Exercises appear at the end of each chapter | |