Probability and Random Processes 4th Edition by Geoffrey R Grimmett, David R Stirzaker ISBN 9780198847595 0198847602

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ISBN 10: 0198847602
ISBN 13: 9780198847595
Author: Geoffrey R Grimmett, David R Stirzaker

Probability and Random Processes 4th Edition Table of 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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Tags: Geoffrey R Grimmett, David R Stirzaker, Probability, Random Processes