PROBABILITY AND RANDOM PROCESSES :

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Cover Probability and Random Processes Copyright Epigraph Preface to the Fourth Edition Contents 1 Events and their probabilities 1.1 Introduction 1.2 Events as sets 1.3 Probability 1.4 Conditional probability 1.5 Independence 1.6 Completeness and product spaces 1.7 Worked examples 1.8 Problems 2 Random variables and their distributions 2.1 Random variables 2.2 The law of averages 2.3 Discrete and continuous variables 2.4 Worked examples 2.5 Random vectors 2.6 Monte Carlo simulation 2.7 Problems 3 Discrete random variables 3.1 Probability mass functions 3.2 Independence 3.3 Expectation 3.4 Indicators and matching 3.5 Examples of discrete variables 3.6 Dependence 3.7 Conditional distributions and conditional expectation 3.8 Sums of random variables 3.9 Simple random walk 3.10 Random walk: counting sample paths 3.11 Problems 4 Continuous random variables 4.1 Probability density functions 4.2 Independence 4.3 Expectation 4.4 Examples of continuous variables 4.5 Dependence 4.6 Conditional distributions and conditional expectation 4.7 Functions of random variables 4.8 Sums of random variables 4.9 Multivariate normal distribution 4.10 Distributions arising from the normal distribution 4.11 Sampling from a distribution 4.12 Coupling and Poisson approximation 4.13 Geometrical probability 4.14 Problems 5 Generating functions and their applications 5.1 Generating functions 5.2 Some applications 5.3 Random walk 5.4 Branching processes 5.5 Age-dependent branching processes 5.6 Expectation revisited 5.7 Characteristic functions 5.8 Examples of characteristic functions 5.9 Inversion and continuity theorems 5.10 Two limit theorems 5.11 Large deviations 5.12 Problems 6 Markov chains 6.1 Markov processes 6.2 Classification of states 6.3 Classification of chains 6.4 Stationary distributions and the limit theorem 6.5 Reversibility 6.6 Chains with finitely many states 6.7 Branching processes revisited 6.8 Birth processes and the Poisson process 6.9 Continuous-time Markov chains 6.10 Kolmogorov equations and the limit theorem 6.11 Birth–death processes and imbedding 6.12 Special processes 6.13 Spatial Poisson processes 6.14 Markov chain Monte Carlo 6.15 Problems 7 Convergence of random variables 7.1 Introduction 7.2 Modes of convergence 7.3 Some ancillary results 7.4 Laws of large numbers 7.5 The strong law 7.6 The law of the iterated logarithm 7.7 Martingales 7.8 Martingale convergence theorem 7.9 Prediction and conditional expectation 7.10 Uniform integrability 7.11 Problems 8 Random processes 8.1 Introduction 8.2 Stationary processes 8.3 Renewal processes 8.4 Queues 8.5 The Wiener process 8.6 L´evy processes and subordinators 8.7 Self-similarity and stability 8.8 Time changes 8.9 Existence of processes 8.10 Problems 9 Stationary processes 9.1 Introduction 9.2 Linear prediction 9.3 Autocovariances and spectra 9.4 Stochastic integration and the spectral representation 9.5 The ergodic theorem 9.6 Gaussian processes 9.7 Problems 10 Renewals 10.1 The renewal equation 10.2 Limit theorems 10.3 Excess life 10.4 Applications 10.5 Renewal–reward processes 10.6 Problems 11 Queues 11.1 Single-server queues 11.2 M/M/1 11.3 M/G/1 11.4 G/M/1 11.5 G/G/1 11.6 Heavy traffic 11.7 Networks of queues 11.8 Problems 12 Martingales 12.1 Introduction 12.2 Martingale differences and Hoeffding’s inequality 12.3 Crossings and convergence 12.4 Stopping times 12.5 Optional stopping 12.6 The maximal inequality 12.7 Backward martingales and continuous-time martingales 12.8 Some examples 12.9 Problems 13 Diffusion processes 13.1 Introduction 13.2 Brownian motion 13.3 Diffusion processes 13.4 First passage times 13.5 Barriers 13.6 Excursions and the Brownian bridge 13.7 Stochastic calculus 13.8 The Itˆo integral 13.9 Itˆo’s formula 13.10 Option pricing 13.11 Passage probabilities and potentials 13.12 Problems Appendix I Foundations and notation (A) Basic notation (B) Sets and counting (C) Vectors and matrices (D) Convergence (E) Complex analysis (F) Transforms (G) Difference equations (H) Partial differential equations Appendix II Further reading Appendix III History and varieties of probability History Varieties Appendix IV John Arbuthnot’s Preface to Of the laws of chance (1692) Appendix V Table of distributions Appendix VI Chronology Bibliography Notation Index