Timeline for Joint irreducibility and aperiodicity of two independent Markov chains
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2022 at 7:34 | history | bounty ended | Dasherman | ||
| Nov 20, 2022 at 14:28 | comment | added | Iosif Pinelis | @Dasherman : Unfortunately, I don't know how to complete the proof of the $\psi$-irreducibility or to find a counterexample. I have posted a question about a special case of this at mathoverflow.net/questions/434854/… but have not received a valid answer yet. It appears that, if a counterexample exists, it should be rather exotic. | |
| Nov 20, 2022 at 9:49 | comment | added | Dasherman | I see, thank you. So to summarize: you've shown that the product chain is aperiodic, but the proof for $\psi$-irreducibility isn't complete yet. Do you think the claim is true without additional conditions, or do you perhaps have (a path towards) a counterexample in mind? | |
| Nov 18, 2022 at 4:26 | comment | added | Iosif Pinelis | @Dasherman : By Theorem 5.4.4 in the book, this answer does prove that the product chain is aperiodic -- because in that theorem one can use any $\nu_k$-small set $C\in\mathscr B^+(\mathcal X)$ with $\nu_k(C)>0$. As for the $\psi$-irreducibility, I have added an idea on how to extend its limited version to the complete one. | |
| Nov 18, 2022 at 4:24 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 880 characters in body
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| Nov 18, 2022 at 0:56 | comment | added | Dasherman | Thanks for the detailed answer! Why is it sufficient to only consider rectangles of positive measure? It is not immediately clear to me why this is sufficient for irreducibility nor for aperiodicity. There may be sets of positive measure in the product space that don't contain any rectangles of positive measure. | |
| Nov 17, 2022 at 23:16 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |