Timeline for Conditional distribution of steps of random walk given the sum
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 26 at 8:33 | comment | added | user8965 | In general that should follow from the strong ratio limit theorem, see Section 4.3 in arxiv.org/abs/1511.01721 for a recent reference. | |
| Nov 25 at 22:32 | history | edited | Viktor B | CC BY-SA 4.0 |
Forgot limit n to ininity
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| Oct 9, 2022 at 0:06 | comment | added | Iosif Pinelis | The real challenge here is to show that this holds for all zero-mean $\mu$ (or maybe to construct a counterexample). | |
| Oct 6, 2022 at 21:35 | comment | added | Viktor B | @IosifPinelis Indeed, $\mathbb{P}\{|S_n| \leq a \} > 0$ is required at least for large $n$. If $S_n$ is non-lattice, then $\mathbb{P}\{|S_n| \leq a \} > 0$ follows from the local limit theorem, whereas if $S_n$ is lattice, it has to be aperiodic (or (1) has to be adapted). The local limit theorem holds for a distribution in the domain of attraction of a stable distribution, so it seems to me that at least for those distributions the proof (1) can follow pretty much along the same lines as in the post. | |
| Oct 6, 2022 at 20:36 | comment | added | Iosif Pinelis | You also have to assume that $P(|S_n|\le a)\ne0$. Then, I think, this should be true for all zero-mean $\mu$ -- but the proof is probably quite nontrivial. | |
| Oct 6, 2022 at 20:21 | history | edited | Viktor B | CC BY-SA 4.0 |
Added proof idea based on Thomas Kojar's comment
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| Oct 6, 2022 at 20:01 | comment | added | Viktor B | I think you are correct and for a wide class of $\mu$ (1) is going to follow a local limit theorem. I modify the post. | |
| Oct 6, 2022 at 19:28 | comment | added | Thomas Kojar | How about doing it by hand by writing $$S_{n+1}=X_1+R_n$$ and then applying strong law to show that the effect of X1 is negligible in the second event in $$P(X_1>t, -a/n<X_1/n+R_n/n<a/n)\approx P(X_1>t) P( -a/n-c_n<R_n/n<a/n+c_n)+o(g_n)$$ for some $g_n,c_n\to 0$ at least when if we assume that X1 is upper/lower bounded? | |
| Oct 6, 2022 at 18:35 | history | asked | Viktor B | CC BY-SA 4.0 |