Timeline for Uniform distribution mod $1$ vs independence of random variables
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 24, 2022 at 6:21 | comment | added | asrxiiviii | Great thanks, I'll check these out asap. | |
| Mar 23, 2022 at 22:19 | comment | added | Steven Stadnicki | More references can be found by searching; the best magic words I've found are 'Discrete Kronecker Weil Convergence', which turn up for instance this paper that seems likely to have some of the answers you're looking for. | |
| Mar 23, 2022 at 22:19 | comment | added | Steven Stadnicki | Apologies — for some reason I'd gotten it in my head that your formula for multidimensional uniform distribution had an error bound in it, but it's just a statement on the limit and not the rate of approach. It appears that explicit rates of convergence are hard to come by and depend on some very fiddly irrationality properties of the specific $a_i$; see e.g. mathoverflow.net/questions/162875/…. | |
| Mar 23, 2022 at 21:02 | comment | added | asrxiiviii | @Steven Stadnicki. Did you mean for each individual $X_j$? If so, I already gave the explicit formula for the CDF of $X_j$, so the bound follows immediately right? (It is $O(1/T)$ as far as I can see.) But what info do I gain about the joint CDF of $(X_1, \cdots, X_k)$ just from this bound? | |
| Mar 23, 2022 at 19:57 | comment | added | Steven Stadnicki | Have you tried starting by bounding the distance between the CDF of $X_j$ and the uniform distribution? | |
| Mar 23, 2022 at 18:05 | history | asked | asrxiiviii | CC BY-SA 4.0 |