Stochastic Simulation

Contents

1   Aims of Simulation

1.1   The Tools

1.2   Models

1.3   Simulation as Experimentation

1.4   Simulation in Inference

1.5   Examples

1.6   Literature

1.7   Conventions

Exercises

2   Pseudo-Random Numbers

2.1   History and Philosophy

2.2   Congruential Generators

2.3   Shift Register Generators

2.4   Lattice Structure

2.5   Shuffling and Testing

2.6   Conclusions

2.7   Proofs

Exercises

3   Random Variables

3.1   Simple Examples

3.2   General Principles

3.3   Discrete Distributions

3.4   Continuous Distributions

3.5   Recommendations

Exercises

4   Stochastic Models

4.1   Order Statistics

4.2   Multivariate Distributions

4.3   Poisson Processes and Lifetimes

4.4   Markov Processes

4.5   Gaussian Processes

4.6   Point Processes

4.7   Metropolis’ Method and Random Fields

Exercises

5   Variance Reduction

5.1   Monte-Carlo Integration

5.2   Importance Sampling

5.3   Control and Antithetic Variates

5.4   Conditioning

5.5   Experimental Design

Exercises

6   Output Analysis

6.1   The Initial Transient

6.2   Batching

6.3   Time-Series Methods

6.4   Regenerative Simulation

6.5   A Case Study

Exercises

7   Uses of Simulation

7.1   Statistical Inference

7.2   Stochastic Methods in Optimization

7.3   Systems of Linear Equations

7.4   Quasi-Monte-Carlo Integration

7.5   Sharpening Buffon’s Needle

Exercises

References

Appendix A   Computer Systems

Appendix B   Computer Programs

B.1   Form a × b mod c

B.2   Check Primitive Roots

B.3   Lattice Constants for Congruential Generators

B.4   Testing ...