# The deviation matrix of a continuous-time Markov chain

Pauline Coolen-Schrijner, Erik A. van Doorn

Research output: Contribution to journalArticleAcademicpeer-review

46 Citations (Scopus)

## Abstract

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 language English 351-366 15 Probability in the engineering and informational sciences 16 3 https://doi.org/10.1017/S0269964802163066 Published - 2002

## Keywords

• Birth-death process
• ergodic Markov chain
• deviation matrix
• convergence to stationarity
• IR-62339
• EWI-12837
• METIS-206668