Timeline for Ergodicity of linear dynamical systems and convergence of covariance matrices
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2023 at 0:40 | comment | added | JGWang | In my post, the $T$ defined by (R1) is ergodic, than for all measurable(and integrable) functional, $\pi_t$ of $\xi$, $(1/n)\sum_{k=0}^{n-1}\pi_t(T^k(\xi))\to\int\pi_t(\xi)d\mu$ is ture. By the way, if you have other problem, you could give a new post, because it is not convenient discussion here. | |
| Aug 21, 2023 at 0:32 | comment | added | JGWang | In my post, the $T$ defined by (R1) is ergodic, than for all measurable(and integrable) functional, $\pi_t$ of $\xi$, $(1/n)\sum_{k=0}^{n-1}\pi_t(T^k(\xi))\to\int\pi_t(\xi)d\mu$. | |
| Aug 20, 2023 at 10:40 | comment | added | Augusto Santos | Just to confirm: The idea would be to resort to Birkhoff's Theorem to establish convergence, correct? Remark that $T$ maps sample paths into sample paths (endowed with the product Lebesgue). If we choose $\pi_t$ to be the projection map, it returns the value of the process at time $t$, then Birkhoff's would imply $(1/n)\sum_{k=0}^{n-1} \pi_t(T^k (w)) \overset{a.s.}\longrightarrow \int \pi_t d\mu = 0$, where $w$ is a centered process whose distribution is kept invariant under the shift $T$ and $\mu$ is Lebesgue. Is that the idea? | |
| Aug 20, 2023 at 9:51 | comment | added | JGWang | Thank you for your upvote. Also, you are right, map $T$ is ergodic. | |
| Aug 20, 2023 at 8:49 | comment | added | Augusto Santos | I do not have the book with me, but I believe the map $T$ that you are referring to is the (one-lag) time shift map (which is ergodic, indeed)? I see, let me think about (+1 upvote). | |
| Aug 20, 2023 at 1:46 | comment | added | JGWang | @AugustoSantos Please see the added Remark. | |
| Aug 20, 2023 at 1:44 | history | edited | JGWang | CC BY-SA 4.0 |
Add Remak
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| Aug 19, 2023 at 4:06 | comment | added | Augusto Santos | JGWang, why is $(y(i))_{i\in\mathbb{N}}$ ergodic? | |
| Aug 19, 2023 at 1:01 | history | edited | JGWang | CC BY-SA 4.0 |
added 1 character in body
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| Aug 19, 2023 at 0:54 | comment | added | JGWang | @AugustoSantos Thank you for your reply. I rewrite last several lines. Welcome to propose any question about this post. | |
| Aug 19, 2023 at 0:43 | history | edited | JGWang | CC BY-SA 4.0 |
rewrite the last several lines.
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| Aug 18, 2023 at 11:19 | comment | added | Augusto Santos | JGWang, thank you. Yes, the process is ergodic (and its sample mean converge almost surely). More concretely, my question is how to prove (or specific reference) that the empirical lag-moments converge almost surely to the limit covariances. Thanks again. | |
| Aug 18, 2023 at 9:06 | history | answered | JGWang | CC BY-SA 4.0 |