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数学与物理
理论数学
Vol. 11 No. 12 (December 2021)
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早期停止定理在随机游动中的应用
The Application of the Early Stopping Theorem in Random Walk
DOI:
10.12677/PM.2021.1112226
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作者:
赵 盼
:北京联合大学,数理与交叉科学研究院,北京
关键词:
随机游动
;
早期停止定理
;
强平稳对偶
;
随机控制
;
随机单调
;
Random Walk
;
The Early Stopping Theorem
;
Strong Stationary Duality
;
Stochastically Dominate
;
Stochastically Monotone
摘要:
Diaconis 和 Fill 利用早期停止定理, 给出ℤ
+
上的随机游动收敛到平稳分布的速度估计时出现了错误,本文不仅纠正了这个错误,而且利用 Markov 不等式和早期停止定理,也给出了ℤ
+
上的随机游动收敛到平稳分布的速度估计。
Abstract:
By using the early stopping theorem, Diaconis and Fill made a mistake when dealing with the convergence to stationarity for a random walk. In the paper, we not only correct the mistake, but also give the speed estimation of convergence to stationarity for the random walk, by using the Markov inequation and the early stopping theorem.
文章引用:
赵盼. 早期停止定理在随机游动中的应用[J]. 理论数学, 2021, 11(12): 2023-2030.
https://doi.org/10.12677/PM.2021.1112226
参考文献
[1]
Diaconis, P. and Fill, J.A. (1990) Strong Stationary Times via a New Form of Duality. The Annals of Probability, 18, 1483-1522.
https://doi.org/10.1214/aop/1176990628
[2]
Diaconis, P. and Miclo, L. (2009) On Times to Quasi-Stationarity for Birth and Death Processes. Journal of Theoretical Probability, 22, 558-586.
https://doi.org/10.1007/s10959-009-0234-6
[3]
Diaconis, P. and Saloff-Coste, L. (2006) Separation Cut-Offs for Birth and Death Chains. The Annals of Applied Probability, 16, 2098-2122.
https://doi.org/10.1214/105051606000000501
[4]
Fill, J.A. (2009) The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof. Journal of Theoretical Probability, 22, 543-557.
https://doi.org/10.1007/s10959-009-0235-5
[5]
Fill, J.A. and Kahn, J. (2013) Comparison Inequalities and Fastest-Mixing Markov Chains. The Annals of Applied Probability, 23, 1778-1816.
https://doi.org/10.1214/12-AAP886
[6]
Diaconis, P. and Fill, J.A. (1990) Examples for the Theory of Strong Stationary Duality with Countable State Spaces. Probability in the Engineering and Informational Sciences, 4, 157-180.
https://doi.org/10.1017/S0269964800001522
[7]
白晶晶, 李培森, 张余辉, 赵盼. 离散时间单生过程的判别准则[J]. 北京师范大学学报(自然科学版), 2015, 51(3): 227-235.
[8]
Shiganov, I.S. (1986) Refinement of the Upper Bound of the Constant in the Central Limit Theorem. Journal of Soviet Mathematics, 35, 2545-2550.
https://doi.org/10.1007/BF01121471
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