Continuous-Time Markov Chains

Chapter
Part of the Springer Texts in Statistics book series (STS, volume 0)

Abstract

This chapter introduces the subject of continuous-time Markov chains [23, 52, 59, 80, 106, 107, 118, 152]. In practice, continuous-time chains are more useful than discrete-time chains. For one thing, continuous-time chains permit variation in the waiting times for transitions between neighboring states. For another, they avoid the annoyances of periodic behavior. Balanced against these advantages is the disadvantage of a more complex theory involving linear differential equations. The primary distinction between the two types of chains is the substitution of transition intensities for transition probabilities. Once one grasps this difference, it is straightforward to formulate relevant continuous-time models. Implementing such models numerically and understanding them theoretically then require the matrix exponential function. Kendall’s birth-death-immigration process, treated at the end of the chapter, involves an infinite number of states and transition intensities that depend on time.

Keywords

Attenuation Covariance Adenine Pyrimidine Purine 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departments of Biomathematics, Human Genetics, and StatisticsUniversity of California, Los AngelesLos AngelesUSA

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