# Book:Geoffrey Grimmett/Probability and Random Processes/Third Edition

## Geoffrey Grimmett and David Stirzaker: *Probability and Random Processes (3rd Edition)*

Published $2001$, **Oxford University Press**

- ISBN 978-0198572220.

### Subject Matter

### 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 revised
- 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 Uniform semigroups
- 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 Existence of processes
- 8.7 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 Itô integral
- 13.9 Itô's formula
- 13.10 Option pricing
- 13.11 Passage probabilities and potentials
- 13.12 Problems

**Appendix I.**Foundations and notation**Appendix II.**Further reading**Appendix III.**History and varieties of probability**Appendix IV.**John Arbuthnot's Preface to*Of the laws of chance*(1692)**Appendix V.**Table of distributions**Appendix VI.**Chronology**Bibliography****Notation****Index**