# The deviation matrix of a continuous-time Markov chain

Pauline Coolen-Schrijner, Erik A. van Doorn

42 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

### Fingerprint

Continuous-time Markov Chain
Markov processes
Deviation
Pi
Birth-death Process
Transition Probability Matrix
Stationarity
Continuous-time Markov chain
Markov chain

### Keywords

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

### Cite this

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title = "The deviation matrix of a continuous-time Markov chain",
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.",
keywords = "Birth-death process, ergodic Markov chain, deviation matrix, convergence to stationarity, IR-62339, EWI-12837, METIS-206668",
author = "Pauline Coolen-Schrijner and {van Doorn}, {Erik A.}",
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language = "English",
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pages = "351--366",
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The deviation matrix of a continuous-time Markov chain. / Coolen-Schrijner, Pauline; van Doorn, Erik A.

In: Probability in the engineering and informational sciences, Vol. 16, No. 3, 2002, p. 351-366.

TY - JOUR

T1 - The deviation matrix of a continuous-time Markov chain

AU - Coolen-Schrijner, Pauline

AU - van Doorn, Erik A.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Birth-death process

KW - ergodic Markov chain

KW - deviation matrix

KW - convergence to stationarity

KW - IR-62339

KW - EWI-12837

KW - METIS-206668

U2 - 10.1017/S0269964802163066

DO - 10.1017/S0269964802163066

M3 - Article

VL - 16

SP - 351

EP - 366

JO - Probability in the engineering and informational sciences

JF - Probability in the engineering and informational sciences

SN - 0269-9648

IS - 3

ER -