|
|
马尔可夫链次几何遍历性的等价条件及其在M/G/1排队中的应用
|
Abstract:
马尔可夫链的遍历性研究在随机过程理论中占有重要地位，本文聚焦于马尔可夫链的次几何遍历性，提出了次几何遍历的六个等价判定条件，这些等价条件对进一步探索一般马尔可夫过程的次几何遍历性提供了理论基础，并将其应用于排队论中经典的M/G/1嵌入过程，得到了易验证的M/G/1嵌入排队过程次几何遍历的判定条件。
The study of the ergodicity of Markov chains holds a significant position in the theory of stochastic processes. This paper focuses on the subgeometric ergodicity of Markov chains and proposes six equivalent criteria for subgeometric ergodicity of Markov chains. These equivalent conditions provide a theoretical foundation for further exploration of the subgeometric ergodicity of general Markov processes. In addition, this paper applies these criteria to the classical M/G/1 embedded process in queueing theory, obtaining easily verifiable conditions for determining the subgeometric ergodicity of the M/G/1 embedded queueing process.
| [1] | Douc, R., Moulines, E. and Soulier, P. (2007) Computable Convergence Rates for Sub-Geometric Ergodic Markov Chains. Bernoulli, 13, 831-848. https://doi.org/10.3150/07-bej5162 |
| [2] | Liu, Y., Zhang, H. and Zhao, Y. (2010) Subgeometric Ergodicity for Continuous-Time Markov Chains. Journal of Mathematical Analysis and Applications, 368, 178-189. https://doi.org/10.1016/j.jmaa.2010.03.019 |
| [3] | Deng, C. (2020) Subgeometric Rates of Convergence for Discrete-Time Markov Chains under Discrete-Time Subordination. Journal of Theoretical Probability, 33, 522-532. https://doi.org/10.1007/s10959-019-00879-z |
| [4] | Durmus, A., Fort, G. and Moulines, É. (2016) Subgeometric Rates of Convergence in Wasserstein Distance for Markov Chains. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 52, 1799-1822. https://doi.org/10.1214/15-aihp699 |
| [5] | Roberts, G.O. and Rosenthal, J.S. (2023) Polynomial Convergence Rates of Piecewise Deterministic Markov Processes. Methodology and Computing in Applied Probability, 25, Article No. 6. https://doi.org/10.1007/s11009-023-09977-2 |
| [6] | Gallegos-Herrada, M.A., Ledvinka, D. and Rosenthal, J.S. (2024) Equivalences of Geometric Ergodicity of Markov Chains. Journal of Theoretical Probability, 37, 1230-1256. https://doi.org/10.1007/s10959-023-01240-1 |
| [7] | Wihstutz, V. (1997) BOOK REVIEW of “Markov Chains and Stochastic Stability” by S.P. Meyn and R.L. Tweedie. The Annals of Probability, 25,1536-1540. https://doi.org/10.1214/aop/1024404525 |
| [8] | Jarner, S.F. and Roberts, G.O. (2002) Polynomial Convergence Rates of Markov Chains. The Annals of Applied Probability, 12, 224-247. https://doi.org/10.1214/aoap/1015961162 |
| [9] | Fort, G. and Moulines, E. (2003) Polynomial Ergodicity of Markov Transition Kernels. Stochastic Processes and their Applications, 103, 57-99. https://doi.org/10.1016/s0304-4149(02)00182-5 |
