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Contents
Contents
Markov Chains and Monte-Carlo Simulation
University Ulm
Department of Stochastics
Lecture Notes
Prof. Dr. Volker Schmidt
Summer 2006
U
LM,
J
ULY 2006
Contents
Introduction
Markov Chains
Specification of the Model and Examples
State Space, Initial Distribution and Transition Probabilities
Examples
Recursive Representation
The Matrix of the
-Step Transition Probabilities
Ergodicity and Stationarity
Basic Definitions and Quasi-positive Transition Matrices
Estimates for the Rate of Convergence; Perron-Frobenius-Theorem
Irreducible and Aperiodic Markov Chains
Stationary Initial Distributions
Direct and Iterative Computation Methods
Reversibility; Estimates for the Rate of Convergence
Definition and Examples
Recursive Construction of the ,,Past''
Determining the Rate of Convergence under Reversibility
Multiplicative Reversible Version of the Transition Matrix; Spectral Representation
Alternative Estimate for the Rate of Convergence;
Contrast
Dirichlet-Forms and Rayleigh-Theorem
Bounds for the Eigenvalues
and
Monte-Carlo Simulation
Generation of Pseudo-Random Numbers
Simple Applications; Monte-Carlo Estimators
Linear Congruential Generators
Statistical Tests
Transformation of Uniformly Distributed Random Numbers
Inversion Method
Transformation Algorithms for Discrete Distributions
Acceptance-Rejection Method
Quotients of Uniformly Distributed Random Variables
Simulation Methods Based on Markov Chains
Example: Hard-Core Model
Gibbs Sampler
Metropolis-Hastings Algorithm
Error Analysis for MCMC Simulation
Estimate for the Rate of Convergence
MCMC Estimators; Bias and Fundamental Matrix
Asymptotic Variance of Estimation; Mean Squared Error
Coupling Algorithms; Perfect MCMC Simulation
Coupling to the Future; Counterexample
Propp-Wilson Algorithm; Coupling from the Past
Monotone Coupling Algorithms
Examples: Birth-and-Death Processes; Ising Model
Read-Once Modification of the CFTP Algorithm
About this document ...
Ursa Pantle 2006-07-20
Contrast
and
-Step Transition Probabilities