Timeline for Connectedness of random distance graph on integers
Current License: CC BY-SA 2.5
9 events
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| Apr 22, 2010 at 20:22 | comment | added | Did | @domotorp: About recurrence/transience, you might try section 1.5 of the book available at "statslab.cam.ac.uk/~james/Markov". "(Lyapunov-)Foster criteria" is a loose name for a variety of drift conditions ensuring the recurrence or the transience of a given Markov chain; some of them are described at "math.ucsd.edu/~pfitz/downloads/courses/spring05/math280c/…". | |
| Apr 22, 2010 at 19:52 | comment | added | domotorp | I was also referring to this when I wrote that I don't see the equivalence. Could anyone please give a reference to these? They were neither on wikipedia, nor on mathworld, and at other places even the statement was too complicated without knowing a bunch of other things. | |
| Apr 22, 2010 at 19:34 | history | edited | Did | CC BY-SA 2.5 |
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| Apr 22, 2010 at 18:59 | comment | added | Did | @Thorny: You are right about your 1), thanks. Answer edited. | |
| Apr 22, 2010 at 18:22 | history | edited | Did | CC BY-SA 2.5 |
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| Apr 22, 2010 at 11:17 | comment | added | Thorny | 1) You are solving the problem of whether there is a path between 1 and N that consists of two monotone subpaths; this is not exactly the same as whether there is a path between 1 and N. 2) If $\sum_{j<n} p(j)$ grows slower than $log(n)$, the steps of $z$ are not integrable. | |
| Apr 22, 2010 at 8:14 | comment | added | domotorp | I think you follow a similar approach that I suggested, with taking the path I described, except that you do it for two starting points at the same time, this seems to be a better idea. I do not see why x_n=y_n would be equivalent to 1 and N being in the same component, but it is surely sufficient. Unfortunately I am really not an expert and I don't understand the notions that you use after. What is integrable step, recurrent, what does Foster's criterion say? Could you please give a reference where I can find these? | |
| Apr 21, 2010 at 21:29 | history | edited | Did | CC BY-SA 2.5 |
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| Apr 21, 2010 at 21:07 | history | answered | Did | CC BY-SA 2.5 |