Probability in the Engineering and Informational Sciences



THE DEVIATION MATRIX OF A CONTINUOUS-TIME MARKOV CHAIN


Pauline  Coolen-Schrijner  a1 and Erik A.  van Doorn  a2
a1 Department of Mathematical Sciences, University of Durham, Durham, UK, E-mail: [email protected]
a2 Faculty of Mathematical Sciences, University of Twente, Enschede, The Netherlands, E-mail: [email protected]

Abstract

The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P(·) and ergodic matrix [Pi] is the matrix D [identical with] [integral operator]0[infty infinity](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.