Timeline for Asymmetric random walk on the line with barriers
Current License: CC BY-SA 3.0
11 events
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| Jan 30, 2015 at 2:06 | comment | added | Douglas Zare | One simple observation is that if the distribution is supported on $\mathbb{N} \cup \lbrace -1 \rbrace$ then it can't be an example, since there would never be any overshoot in the negative direction yet asymmetric distributions would have some positive overshooting. Similarly, the distribution has to include a positive probability of taking a positive step of size at least $2$. | |
| Jan 29, 2015 at 20:05 | comment | added | Liviu Nicolaescu | It looks to me that you have a one parameter family of random walks, one random walk for every natural number $n$. The random walk $(X_\bullet^n)$ corresponding to $n$ satisfies $X_{k+1}^n=X_k^n$ if $|X_k|\geq n$, and $X_{k+1}^n=X_k+S$ where $S$ is the $\mathbb{Z}$-random variable distributed according to the given (compactly supported) distribution. | |
| Jan 29, 2015 at 18:52 | comment | added | Anthony Quas | Trivial comment: From the optional stopping theorem, if the expected one-step displacement is 0, then asymptotically the probability of ending at each barrier is 1/2. | |
| Jan 29, 2015 at 18:50 | comment | added | Anthony Quas | @Pace: From the sentence that begins "Obviously", I think it's reasonably clear the OP means symmetric in the sense that the distribution of $X_k-X_{k-1}$ is the same as the distribution of $-(X_k-X_{k-1})$. | |
| Jan 29, 2015 at 18:20 | comment | added | Pace Nielsen | Would $P(-2)=1/3$, $P(1)=2/3$ be an asymmetric probability distribution? Or by asymmetry do you mean a distribution where the expected amount of movement on the first step is (say) to the right? | |
| Jan 29, 2015 at 17:59 | comment | added | Timothy Chow | O.K., I edited the description. | |
| Jan 29, 2015 at 17:59 | history | edited | Timothy Chow | CC BY-SA 3.0 |
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| Jan 29, 2015 at 16:48 | comment | added | Liviu Nicolaescu | Could you edit the question so the $n$-s are replaced with the $N$-s in the right places? | |
| Jan 29, 2015 at 16:41 | comment | added | Timothy Chow | Maybe it would have been clearer to use different symbols, but yes, I'm asking about a fixed probability distribution that has compact support, and letting the barriers go to infinity. | |
| Jan 29, 2015 at 15:33 | comment | added | Anthony Quas | It looks like the $n$ in the set-up is not the same as the $n$ ($N$?) in the question, right? i.e. the range of the displacements are in $[-n,n]$, but the barriers are at $\pm N$ | |
| Jan 29, 2015 at 15:10 | history | asked | Timothy Chow | CC BY-SA 3.0 |