The deviation matrix of a continuous-time Markov chain

The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P(.) and ergodic matrix Pi is the matrix Dequivalent tof(o)(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.