Timeline for Expectation of first positive value in random walk
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2010 at 15:54 | history | edited | Mark Meckes | CC BY-SA 2.5 |
Corrected "expectation", added a missing space
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| Oct 29, 2010 at 15:32 | answer | added | Did | timeline score: 2 | |
| Oct 28, 2010 at 14:30 | comment | added | Did | 1) If $X_k\le x$ almost surely, then $0 < S_N\le x$ almost surely, hence $S_N$ is indeed integrable. 2) I don't understand why $S_N$ should take a finite number of values in general (this seems to be the case if and only if $p$ is rational). 3) If $p=1/(k+1)$, then $S_N$ is uniformly distributed on the integers from $1$ to $k$, in particular $E(S_N)=(k+1)/2$. For instance, if $p=1/3$, then $S_N$ is uniform on $1$ and $2$, hence $E(S_N)=3/2$. | |
| Oct 27, 2010 at 17:04 | comment | added | Michael Lugo | OK. I meant that $S_n$ is integer-valued for all $n$, and so $S_N$ always takes positive integer values. | |
| Oct 27, 2010 at 15:13 | answer | added | user6096 | timeline score: 5 | |
| Oct 27, 2010 at 8:24 | comment | added | Ewan Delanoy | @JBL : For $p=\frac{1}{3}$ I can show that the expectancy is greater than 1. | |
| Oct 27, 2010 at 8:23 | comment | added | Ewan Delanoy | @ Shai : corrected the typo, thanks. | |
| Oct 27, 2010 at 8:22 | history | edited | Ewan Delanoy | CC BY-SA 2.5 |
added 476 characters in body
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| Oct 26, 2010 at 16:37 | comment | added | Michael Lugo | JBL: In the case $p = 1/3$, $X_k$ is $-1$ with probability $2/3$ and $2$ with probability $1/3$. So $S_n$ is positive-integer-valued for all $n$. In particular $S_N$ only takes the values $1$ or $2$, and therefore has expectation at least $1$. | |
| Oct 26, 2010 at 15:05 | comment | added | Shai Covo |
There is an obvious typo: if $p$ is of the form $1−1/k$ where $k$ is an integer, then $S_N$ is equal to $1/(k−1)$
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| Oct 26, 2010 at 13:12 | history | asked | Ewan Delanoy | CC BY-SA 2.5 |