Timeline for Random Walk in $\mathbb{R}^n$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2012 at 14:39 | comment | added | Gerald Edgar | Look for some sources ... en.wikipedia.org/wiki/Random_walk | |
| Feb 17, 2012 at 14:18 | comment | added | Richard Stanley | A related result is Rayleigh's theorem: start at the origin in the plane and make $n$ steps of length one, with directions chosen independently and uniformly at each step. Then the probability of ending up at distance less than one from the origin is $1/(n+1)$. For an elegant recent proof by Bernardi, see math.mit.edu/~bernardi/publications/Rayleigh.pdf. | |
| Feb 17, 2012 at 7:55 | comment | added | alexei | That was very helpful. Thank you very much. | |
| Feb 17, 2012 at 6:44 | comment | added | Anthony Quas | In the real case you need to change the definition of recurrence: there's 0 probability of ever hitting any specific point, so you look instead at recurrence to a neighbourhood (i.e. a random walk is recurrent if for each initial point $x$ and for each $\epsilon > 0$, the walk almost surely returns to an $\epsilon$-neighbourhood of $x$. With this definition, the results are the same as in the discrete case: the random walk is recurrent in 1 and 2 dimensions; and transient in 3 dimensions or higher. | |
| Feb 17, 2012 at 4:34 | history | asked | alexei | CC BY-SA 3.0 |