Understanding Markov Chains: Examples and Applications

Introduction 1 Probability Background 1.1 Probability Spaces and Events 1.2 Probability Measures 1.3 Conditional Probabilities and Independence 1.4 Random Variables 1.5 Probability Distributions 1.6 Expectation of a Random Variable 1.7 Conditional Expectation 1.8 Moment and Probability Generating Functions Exercises 2 Gambling Problems 2.1 Constrained Random Walk 2.2 Ruin Probabilities 2.3 Mean Game Duration Exercises 3 Random Walk 3.1 Unrestricted Random Walk 3.2 Mean and Variance 3.3 Distribution 3.4 First Return to Zero Exercises 4 Discrete-Time Markov Chains 4.1 Markov Property 4.2 Transition matrix 4.3 Examples of Markov Chains 4.4 Higher Order Transition Probabilities 4.5 The Two-State Discrete-Time Markov Chain Exercises 5 First Step Analysis 5.1 Hitting Probabilities 5.2 Mean Hitting and Absorption Times 5.3 First Return Times 5.4 Number of Returns Exercises 6 Classication of States 6.1 Communicating States 6.2 Recurrent States 6.3 Transient States 6.4 Positive and Null Recurrence 6.5 Periodicity and Aperiodicity Exercises 7 Long-Run Behavior of Markov Chains 7.1 Limiting Distributions 7.2 Stationary Distributions 7.3 Markov Chain Monte Carlo Exercises 8 Branching Processes 8.1 Defnition and Examples 8.2 Probability Generating Functions 8.3 Extinction Probabilities Exercises 9 Continuous-Time Markov Chains 9.1 The Poisson Process 9.2 Continuous-Time Chains 9.3 Transition Semigroup9.4 Infinitesimal Generator 9.5 The Two-State Continuous-Time Markov Chain 9.6 Limiting and Stationary Distributions 9.7 The Discrete-Time Embedded Chain 9.8 Mean Absorption Time and Probabilities Exercises 10 Discrete-Time Martingales 10.1 Filtrations and Conditional Expectations 10.2 Martingales - Definition and Properties 10.3 Ruin Probabilities 10.4 Mean Game Duration Exercises 11 Spatial Poisson Processes 11.1 Spatial Poisson (1781-1840) Processes 11.2 Poisson Stochastic Integrals 11.3 Transformations of Poisson Measures 11.4 Moments of Poisson Stochastic Integrals 11.5 Deviation Inequalities Exercises 12 Reliability Theory 12.1 Survival Probabilities 12.2 Poisson Process with Time-Dependent Intensity 12.3 Mean Time to Failure Exercises Some Useful Identities Solutions to the Exercises References Index