Understanding Markov Chains

Description

This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions. Front Matter ....Pages i-xvii Probability Background (Nicolas Privault)....Pages 1-37 Gambling Problems (Nicolas Privault)....Pages 39-67 Random Walks (Nicolas Privault)....Pages 69-87 Discrete-Time Markov Chains (Nicolas Privault)....Pages 89-113 First Step Analysis (Nicolas Privault)....Pages 115-145 Classification of States (Nicolas Privault)....Pages 147-162 Long-Run Behavior of Markov Chains (Nicolas Privault)....Pages 163-188 Branching Processes (Nicolas Privault)....Pages 189-209 Continuous-Time Markov Chains (Nicolas Privault)....Pages 211-262 Discrete-Time Martingales (Nicolas Privault)....Pages 263-280 Spatial Poisson Processes (Nicolas Privault)....Pages 281-288 Reliability Theory (Nicolas Privault)....Pages 289-293 Back Matter ....Pages 295-372