Efnisyfirlit

• 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

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