Timeline for Infinite positive matrices with probability eigenvector
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2021 at 16:12 | history | edited | SIB | CC BY-SA 4.0 |
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| Jan 30, 2021 at 15:59 | history | edited | SIB | CC BY-SA 4.0 |
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| Jan 30, 2021 at 15:59 | comment | added | Jochen Glueck | Thank you for your response and for the reference! I found the reference quite interesting, since I never looked at Perron-Frobenius theory from this perspective. (For people who are not familiar with the framework of infinite matrices - like me, for instance -, it might still be worthwhile, though, to note explicitly that Kitchens actually says in Remark 7.1.2 that his terminology is "slightly nonstandard", and that, for instance, his usage of "transient" and "recurrent" is not consistent with the usage of these notions in probability theory, in general.) | |
| Jan 30, 2021 at 4:27 | comment | added | SIB | Yes, it is an equivalent formulation of the question saying that $x \in \ell^1$. | |
| Jan 30, 2021 at 4:25 | comment | added | SIB | There is a long list of references about the Perron-Frobenius theorem for infinite matrices. I use the (standard) terminology from Chapter 7 of the book "Symbolic dynamics" by B. Kitchens. This terminology is similar to that used in Markov chains. | |
| Jan 30, 2021 at 3:38 | comment | added | Jochen Glueck | (iii) Do I understand correctly that your question is actually whether $x$ can be chosen to be in $\ell^1$? | |
| Jan 30, 2021 at 3:38 | comment | added | Jochen Glueck | I have difficulties to follow the question. Could you please clarify the following points? (i) What does aperiodic and recurrent mean for an infinite matrix? In my experience, those notions are typically defined for Markov chains, but not for general infinite matrices. (ii) Which result precisely do you refer to as "the generalized Perron-Frobenius theorem"? There are many infinite-dimensional Perron-Frobenius type results, but all those that I have seen so far are about linear operators on certain classes of Banach spaces, not about infinite matrices. | |
| Jan 30, 2021 at 0:33 | history | asked | SIB | CC BY-SA 4.0 |