Spectral theoretic characterisation of Markov chain convergence
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916957810327552 |
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| author | Davies, Bryn Xiao, Angelica Yu |
| author_facet | Davies, Bryn Xiao, Angelica Yu |
| contents | In this work, we characterise the statistics of Markov chains by constructing an associated sequence of periodic differential operators. Studying the density of states of these operators reveals the absolutely continuous invariant measure of the Markov chain. This approach also leads to a direct proof of convergence to the invariant measure, along with explicit convergence rates. We show how our method can be applied to a class of related Markov chains including the logistic map, the tent map and Chebyshev maps of arbitrary order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15829 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral theoretic characterisation of Markov chain convergence Davies, Bryn Xiao, Angelica Yu Dynamical Systems Classical Analysis and ODEs Probability 37A05, 34L05, 58J51 In this work, we characterise the statistics of Markov chains by constructing an associated sequence of periodic differential operators. Studying the density of states of these operators reveals the absolutely continuous invariant measure of the Markov chain. This approach also leads to a direct proof of convergence to the invariant measure, along with explicit convergence rates. We show how our method can be applied to a class of related Markov chains including the logistic map, the tent map and Chebyshev maps of arbitrary order. |
| title | Spectral theoretic characterisation of Markov chain convergence |
| topic | Dynamical Systems Classical Analysis and ODEs Probability 37A05, 34L05, 58J51 |
| url | https://arxiv.org/abs/2410.15829 |