Abstract
The asymptotic variance is an important criterion to evaluate the performance of Markov chains, especially for the central limit theorems. We give the variational formulas for the asymptotic variance of discrete-time (non-reversible) Markov chains on general state space. The variational formulas provide many applications, extending the classical Peskun’s comparison theorem to non-reversible Markov chains, and obtaining several comparison theorems between Markov chains with various perturbations.
Funding Statement
Lu-Jing Huang acknowledges support from NSFC (No. 11901096), NSF-Fujian (No. 2020J05036), the Program for Probability and Statistics: Theory and Application (No. IRTL1704), and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). Yong-Hua Mao acknowledges support from NSFC (No. 11771047) and National Key Research and Development Program of China (2020YFA0712901).
Acknowledgements
We thank the anonymous referees for their careful reading and corrections. Lu-Jing Huang would like to thank Professors Chii-Ruey Hwang, Ting-Li Chen and Dr. Michael C.H. Choi for helpful discussions (Example 4.6 is from Professor Ting-Li Chen).
Citation
Lu-Jing Huang. Yong-Hua Mao. "Variational formulas for asymptotic variance of general discrete-time Markov chains." Bernoulli 29 (1) 300 - 322, February 2023. https://doi.org/10.3150/21-BEJ1458
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