The fourth edition of this successful text provides an introduction to probability and random processes, with many practical applications. It is aimed at mathematics undergraduates and postgraduates, and has four main aims.
US BL To provide a thorough but straightforward account of basic probability theory, giving the reader a natural feel for the subject unburdened by oppressive technicalities. BE BL To discuss important random processes in depth with many examples.BE BL To cover a range of topics that are significant and interesting but less routine.BE BL To impart to the beginner some flavour of advanced work.BE UE
OP The book begins with the basic ideas common to most undergraduate courses in mathematics, statistics, and science. It ends with material usually found at graduate level, for example, Markov processes, (including Markov chain Monte Carlo), martingales, queues, diffusions, (including stochastic calculus with Itô's formula), renewals, stationary processes (including the ergodic theorem), and option pricing in mathematical finance using the Black-Scholes formula. Further, in this new
revised fourth edition, there are sections on coupling from the past, Lévy processes, self-similarity and stability, time changes, and the holding-time/jump-chain construction of continuous-time Markov chains. Finally, the number of exercises and problems has been increased by around 300 to a total of about
1300, and many of the existing exercises have been refreshed by additional parts. The solutions to these exercises and problems can be found in the companion volume, One Thousand Exercises in Probability, third edition, (OUP 2020).CP
Author(s): GEOFFREY STIRZAKER DAVID GRIMMETT
Edition: 4
Publisher: OXFORD UNIV PRESS
Year: 2020
Language: English
City: S.l.
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
