Summary
For a continuous-time finite state Markov process with stationary distribution π, it is well-known thatP i(X t=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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Research supported by National Science Foundation Grant MCS 84-03239
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Aldous, D.J. Finite-time implications of relaxation times for stochastically monotone processes. Probab. Th. Rel. Fields 77, 137–145 (1988). https://doi.org/10.1007/BF01848136
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DOI: https://doi.org/10.1007/BF01848136
Keywords
- Relaxation Time
- Stochastic Process
- Probability Theory
- Markov Process
- Stationary Distribution