Stirzaker G., Grimmett D. Probability and Random Processes

4th edition. — Oxford: Oxford University Press, 2020. — 682 p.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 functions.
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 functions.
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.
Kolmogorov equations and the limit theorem.
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.
L´evy processes and subordinators.
Self-similarity and stability.
Time changes.
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/.
M/G/.
G/M/.
G/G/.
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 Itˆo integral.
Itˆo’s formula.
Option pricing.
Passage probabilities and potentials.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.
Further reading.
History and varieties of probability.
History.
Varieties.
John Arbuthnot’s Preface to Of the laws of chance (1692).
Table of distributions.
Chronology.