The deviation matrix of a continuous-time Markov chain

Pauline Coolen-Schrijner, Erik A. van Doorn

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    he deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix $P(.)$ and ergodic matrix $\Pi$ is the matrix $D \equiv \int_0^{\infty} (P(t)-\Pi)dt$. We give conditions for $D$ to exist and discuss properties and a representation of $D$. The deviation matrix of a birth-death process is investigated in detail. We also describe a new application of deviation matrices by showing that a measure for the convergence to stationarity of a stochastically increasing Markov chain can be expressed in terms of the elements of the deviation matrix of the chain.
    Original languageEnglish
    Pages (from-to)351-366
    Number of pages15
    JournalProbability in the engineering and informational sciences
    Issue number3
    Publication statusPublished - 2002


    • Birth-death process
    • Ergodic Markov chain
    • Deviation matrix
    • Convergence to stationarity


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    • The deviation matrix of a continuous-time Markov chain

      Coolen-Schrijner, P. & van Doorn, E. A., 2001, Enschede: University of Twente. 20 p. (Memorandum / Department of Applied Mathematics; no. 1567)

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