Timeline for A Claim on Typical Voronoi Cells
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 21, 2013 at 6:34 | comment | added | Yoav Kallus | OK, I see that this is indeed a problem when considering an infinite volume. | |
| Dec 20, 2013 at 21:36 | vote | accept | MLT | ||
| Dec 20, 2013 at 21:22 | comment | added | Yuri Bakhtin | @MLT: I don't suffer from lack of publications, and this is hardly a way to advertise your project, so I am skeptical. If you think though your project is interesting, email me. | |
| Dec 20, 2013 at 15:54 | comment | added | Yuri Bakhtin | @YoavKallus: your definition implies that $(x_i)_{i\ge 1}$ is a stationary process. This is bad. For instance, this will mean that there will be finite areas containing infinitely many points. | |
| Dec 20, 2013 at 15:27 | comment | added | Yoav Kallus | @YuriBakhtin: I mean that I have a probability measure $\mu$ on the space of point configurations $(x_1, x_2, \ldots)$, and the measure is invariant under the operation $\pi_{ij}: (x_1, x_2, \ldots x_i, \ldots x_j, \ldots) \mapsto (x_1, x_2, \ldots x_j, \ldots x_i, \ldots)$. That is, if $A$ is a measurable set of configurations, then $\mu(A) = \mu(\pi_{ij}(A))$. | |
| Dec 20, 2013 at 14:58 | comment | added | Yuri Bakhtin | @YoavKallus: If you try to define things precisely you will be in trouble or end up with meaningless statements. For example, what exactly do you mean by "the process is symmetric under relabelling"? In the construction of my answer I start with an arbitrary Poisson configuration and "relabel" it obtaining a labeling that violates the claim. | |
| Dec 20, 2013 at 5:20 | comment | added | Yoav Kallus | To be clear, my comments refer to the case where the the process is symmetric under relabelling. | |
| Dec 20, 2013 at 3:08 | comment | added | Yuri Bakhtin | I am afraid that my example shows that his claim is wrong. | |
| Dec 20, 2013 at 2:38 | comment | added | Yuri Bakhtin | Answering your questions: 1)yes, this is a counterexample. 2) You want to average over all labelings, but on the plane there are infinitely many Poissonian points and infinitely many labelings on them, and there is no natural distribution on them. Of course, you can take a limit over boxes growing to infinity, then you will arrive at some notion of expected average cell size. Comparing individual points in this scheme will not work, their individuality will be lost in the averaging procedure. | |
| Dec 20, 2013 at 2:27 | history | edited | Yuri Bakhtin | CC BY-SA 3.0 |
added 112 characters in body
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| Dec 20, 2013 at 2:25 | comment | added | Yuri Bakhtin | If your point process is on a torus and not the entire plane, then you can do random labeling (uniform distribution on all labelings). However, (1)this applies only to the finite-volume situation, (2)you need an additional source of randomness, (3)points sort of lose their individuality. So I edited my answer slightly. | |
| Dec 20, 2013 at 2:21 | history | edited | Yuri Bakhtin | CC BY-SA 3.0 |
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| Dec 19, 2013 at 23:52 | history | answered | Yuri Bakhtin | CC BY-SA 3.0 |