Grimmett G.R., Stirzaker D.R. Probability and Random Processes

Third Edition. — Oxford University Press, 2001. — 608 p. — ISBN: 0198572239, 978-0198572220.This book gives an introduction to probability and its many practical application by providing a thorough, entertaining account of basic probability and important random processes, covering a range of important topics. Emphasis is on modelling rather than abstraction and there are new sections on sampling and Markov chain Monte Carlo, renewal-reward, queueing networks, stochastic calculus, and option pricing in the Black-Scholes model for financial markets. In addition, there are almost 400 exercises and problems relevant to the material. Solutions can be found in One Thousand Exercises in Probability.Events and their probabilities.
Events as sets, Probability, Conditional probability, Independence, Completeness and product spaces,
Worked examples.
Random variables and their distributions.
Random variables, The law of averages, Discrete and continuous variables, Worked examples,
Random vectors, Monte Carlo simulation.
Discrete random variables.
Probability mass unctions, Independence, Expectation, Indicators and matching, Examples of discrete variables, Dependence, Conditional distributions and conditional expectation, Sums of random variables, Simple random walk, Random walk: counting sample paths.
Continuous random variables.
Probability density unctions, Independence, Expectation, Examples of continuous variables, Dependence, Conditional distributions and conditional expectation, Functions of random variables,
Sums of random variables, Multivariate normal distribution,
Distributions arising from the normal distribution, Sampling from a distribution,
Coupling and Poisson approximation, Geometrical probability.
Generating functions and their applications.
Generating functions, Some applications, Random walk, Branching processes, Age-dependent branching processes, Expectation revisited,
Characteristic functions, Examples of characteristic functions, Inversion and continuity theorems, Two limit theorems, Large deviations.
Markov chains.
Markov processes, Classification of states, Classification of chains, Stationary distributions and the limit theorem, Reversibility,
Chains with finitely many states, Branching processes revisited, Birth processes and the Poisson process,
Continuous-time Markov chains, Uniform semigroups, Birth-death processes and imbedding, Special processes,
Spatial Poisson processes, Markov chain Monte Carlo.
Convergence of random variables.
Modes of convergence, Some ancillary results, Laws of large numbers, The strong law,
The law of the iterated logarithm, Martingales, Martingale convergence theorem, Prediction and conditional expectation, Uniform integrability.
Random processes.
Stationary processes, Renewal processes, Queues, The Wiener process, Existence of processes.
Stationary processes.
Linear prediction, Autocovariances and spectra, Stochastic integration and the spectral representation, The ergodic theorem, Gaussian processes.
Renewals.
The renewal equation, Limit theorems, Excess life, Applications, Renewal-reward processes.
Queues.
Single-server queues, M/M/1, M/G/1, G/M/1, G/G/1, Heavy traffic, Networks of queues.
Martingales.
Martingale differences and Hoeffding's inequality, Crossings and convergence, Stopping times, Optional stopping,
The maximal inequality, Backward martingales and continuous-time martingales, Some examples.
Diffusion processes.
Brownian motion, Diffusion processes, First passage times, Barriers, Excursions and the Brownian bridge,
Stochastic calculus, The Ito integral, Ito's formula, Option pricing, Passage probabilities and potentials.
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.