# Monte Carlo theory, methods and examples

I have a book in progress on Monte Carlo, quasi-Monte Carlo and Markov chain Monte Carlo. Several of the chapters are polished enough to place here. I'm interested in comments especially about errors or suggestions for references to include. There's no need to point out busted links (?? in LaTeX) because the computer will catch those for me when it is time to root out the last of them.

@book{mcbook,
author = {Art B. Owen},
year = 2013,
title = {Monte Carlo theory, methods and examples}
}

Contents
1. Introduction
2. Simple Monte Carlo
3. Uniform random numbers
4. Non-uniform random numbers
5. Random vectors and objects
6. Processes
7. Other integration methods
8. Variance reduction
9. Importance sampling
11. Markov chain Monte Carlo
12. Gibbs sampler
14. Sequential Monte Carlo
15. Quasi-Monte Carlo
16. Lattice rules
17. Randomized quasi-Monte Carlo

Chapters 1 and 2

1 Introduction

1. Example: traffic modeling
2. Example: interpoint distances
3. Notation
4. Outline of the book
5. End notes
6. Exercises

2 Simple Monte Carlo

1. Accuracy of simple Monte Carlo
2. Error estimation
3. Safely computing the standard error
4. Estimating probabilities
5. Estimating quantiles
6. Random sample size
7. When Monte Carlo fails
8. Chebychev and Hoeffding intervals
9. End notes
10. Exercises

3 Uniform Random Numbers

1. Random and pseudo-random numbers
2. States, periods, seeds, and streams
3. U(0,1) random variables
4. Inside a random number generator
5. Uniformity measures
6. Statistical tests of random numbers
7. Pairwise independent random numbers
8. End notes
9. Exercises

4 Non-uniform Random Numbers

1. Inverting the CDF
2. Examples of inversion
3. Inversion for the normal distribution
4. Inversion for discrete random variables
5. Numerical inversion
6. Other transformations
7. Acceptance-rejection
8. Gamma random variables
9. Mixtures and automatic generators
10. End notes
11. Exercises

5 Random vectors and objects

1. Generalizations of one-dimensional methods
2. Multivariate normal and t
3. Multinomial
4. Dirichlet
5. Multivariate Poisson and other distributions
6. Copula-marginal sampling
7. Random points on the sphere
8. Random matrices
9. Example: classification error rates
10. Random permutations
11. Sampling without replacement
12. Random graphs
13. End notes
14. Exercises

6 Processes

1. Stochastic process definitions
2. Discrete time random walks
3. Gaussian processes
4. Detailed simulation of Brownian motion
5. Stochastic differential equations
6. Non-Poisson point processes
7. Dirichlet processes
8. Discrete state, continuous time processes
9. End notes
10. Exercises

1. The midpoint rule
2. Simpson's rule
3. Higher order rules
4. Fubini and the curse of dimensionality
5. Hybrids with Monte Carlo
7. End notes
8. Exercises

8 Variance reduction

1. Overview of variance reduction
2. Antithetics
3. Example: expected log return
4. Stratification
5. Example: stratified compound Poisson
6. Common random numbers
7. Conditioning
8. Example: maximum Dirichlet
9. Control variates
10. Moment matching and reweighting
11. End notes
12. Exercises

9 Importance sampling

1. Basic importance sampling
2. Self-normalized importance sampling
3. Importance sampling diagnostics
4. Example: PERT
5. Importance sampling versus acceptance-rejection
6. Exponential tilting
7. Modes and Hessians
8. General variables and stochastic processes
9. Control variates in importance sampling
10. Mixture importance sampling
11. Multiple importance sampling
12. Positivisation
13. What-if simulations
14. End notes
15. Exercises