Timeline for Some references to understand the proof of a theorem about simple random walk on galton watson trees
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21, 2016 at 14:22 | comment | added | user115608 | @R W i'm not sure about understanding some details. | |
| Aug 21, 2016 at 14:20 | comment | added | user115608 | @R W let me tell u what is bothering me.u said: "The original Galton-Watson measure is not stationary because the simple random walk is not stochastically homogeneous with respect to it: the root is an orphan and therefore has fewer neighbours than the offspring."but GW measure is a measure on the space of trees. right? | |
| Aug 21, 2016 at 14:05 | comment | added | R W | The fact that the GW is not stationary is not used in the proof - so let it be an exercise. | |
| Aug 21, 2016 at 13:46 | comment | added | user115608 | @R W ok.and what about GW? | |
| Aug 21, 2016 at 13:45 | comment | added | R W | OK. Stationarity of $\pi$ follows from the fact that the Markov chain is reversible with respect to it. Namely, if one takes $\pi$ as an initial distribution, then the resulting joint distribution $\Pi$ of the chain at the moments 0 and 1 is invariant under the "flip transformation" which exchanges $x_0$ and $x_1$. | |
| Aug 21, 2016 at 13:33 | comment | added | user115608 | @R W i know that u didnt say so.i want to understand what is the stationary measure?and why u say $\pi$ is ,but not the GW | |
| Aug 21, 2016 at 13:23 | comment | added | R W | I never say that the GW measure is stationary - moreover I emphasize that it is not (the first sentence of paragraph 2). It is the measure $\pi$ that is stationary. Both the GW measure and the measure $\pi$ are defined on the space $\mathcal T$ of rooted trees, so that all the trees appearing in their definitions have a distinguished vertex (root) from the very beginning. | |
| Aug 21, 2016 at 12:53 | comment | added | user115608 | @R W and when u say this measure is stationary?when u don't care about which vertex to be the root?sorry for asking too much | |
| Aug 21, 2016 at 12:49 | comment | added | R W | The GW measure on $\mathcal T$ is the distribution of usual Galton-Watson trees (the root being the progenitor). | |
| Aug 21, 2016 at 12:31 | comment | added | user115608 | @R W ok thanks.and what does "Galton-Watson measure" mean exactly?u mean precisely the measure u said in last comment? | |
| Aug 21, 2016 at 12:28 | comment | added | R W | This measure is explicitly described in the 3rd paragraph of my answer. Once again: this is the distribution of the random rooted trees obtained in the following way: take a random GW tree, add an edge to its root, and then grow another independent GW tree from the other end of this edge. | |
| Aug 21, 2016 at 11:35 | comment | added | user115608 | @R W , if it is true,what do you mean by : "The simple random walk on TT has a stationary probability measure $\pi $ naturally related to the original branching process."?i don't know that measure. | |
| Aug 21, 2016 at 11:33 | comment | added | user115608 | @R W i don't understand completely,you say we can consider the space of trees instead of the space of vertices,right? | |
| Aug 20, 2016 at 16:06 | history | answered | R W | CC BY-SA 3.0 |