Finite-time implications of relaxation times for stochastically monotone processes

SummaryFor a continuous-time finite state Markov process with stationary distribution π, it is well-known thatPi(Xt=j)-πj isO(e-λt) ast→∞, for a certain λ. For a stochastically monotone process for which the reversed process is also stochastically monotone, one can obtain bounds valid for allt. Precisely, $$\sum\limits_i {\pi _j \mathop {\max |}\limits_j P_i } (X_t \leqq j) - \pi [0,j]| \leqq 2(\lambda t + 2)$$ exp(-λt). The proof exploits duality for stochastically monotone processes.

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