Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915652176969728 |
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| author | Grünewälder, Steffen Khaleghi, Azadeh |
| author_facet | Grünewälder, Steffen Khaleghi, Azadeh |
| contents | We propose methods to estimate the individual $β$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path $X_0,X_1, \dots,X_n$. Under standard smoothness conditions on the densities, namely, that the joint density of the pair $(X_0,X_m)$ for each $m$ lies in a Besov space $B^s_{1,\infty}(\mathbb R^2)$ for some known $s>0$, we obtain a rate of convergence of order $\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})$ for the expected error of our estimator in this case\footnote{We use $[s]$ to denote the integer part of the decomposition $s=[s]+\{s\}$ of $s \in (0,\infty)$ into an integer term and a {\em strictly positive} remainder term $\{s\} \in (0,1]$.}. We complement this result with a high-probability bound on the estimation error, and further obtain analogues of these bounds in the case where the state-space is finite. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $\mathcal O(\log(n) n^{-1/2})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07296 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes Grünewälder, Steffen Khaleghi, Azadeh Statistics Theory Machine Learning We propose methods to estimate the individual $β$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path $X_0,X_1, \dots,X_n$. Under standard smoothness conditions on the densities, namely, that the joint density of the pair $(X_0,X_m)$ for each $m$ lies in a Besov space $B^s_{1,\infty}(\mathbb R^2)$ for some known $s>0$, we obtain a rate of convergence of order $\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})$ for the expected error of our estimator in this case\footnote{We use $[s]$ to denote the integer part of the decomposition $s=[s]+\{s\}$ of $s \in (0,\infty)$ into an integer term and a {\em strictly positive} remainder term $\{s\} \in (0,1]$.}. We complement this result with a high-probability bound on the estimation error, and further obtain analogues of these bounds in the case where the state-space is finite. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $\mathcal O(\log(n) n^{-1/2})$. |
| title | Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2402.07296 |