The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P(·) and ergodic matrix Π is the matrix D ≡ ∫0∞(P(t) − Π) 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.
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