Timeline for Uniform distribution modulo 1 and probability [closed]
Current License: CC BY-SA 4.0
16 events
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| Aug 25, 2021 at 17:32 | history | closed |
user44191 user44143 Alexey Ustinov Steven Landsburg LeechLattice |
Not suitable for this site | |
| Aug 24, 2021 at 13:35 | comment | added | TheGrandDuke | @user44191 Classical definition of probability is acceptable. However, does the u.d. mod 1 property and the probability in that matter hold for not fixed numbers? | |
| Aug 24, 2021 at 12:47 | comment | added | user44191 | Presumably, $P(\{a_n\} > \frac{1}{n})$ is meant to be something like $\frac{|\{1 \leq i \leq n | a_i > \frac{1}{n}\}|}{n}$. Or do you specifically want to match the term $a_n$ with the denominator of the fraction $\frac{1}{n}$? | |
| Aug 24, 2021 at 12:43 | history | edited | YCor |
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| Aug 24, 2021 at 12:24 | comment | added | TheGrandDuke | @RonniePavlov thank you very much for the comment. I edited the question and tried to clarify the confusion. | |
| Aug 24, 2021 at 12:22 | history | edited | TheGrandDuke | CC BY-SA 4.0 |
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| Aug 23, 2021 at 16:39 | comment | added | Ronnie Pavlov | In case this is the point of confusion: what do you mean by "probability that $a_n > \epsilon$" here? A sequence doesn't inherently come with a probability! Generally, one would associate the Lebesgue measure, since it's the limit of the empirical measures $\frac{1}{n} \sum_{i=0}^{n-1} \delta_{a_n}$ when $(a_n)$ is uniformly distributed. Then, indeed, the answer is trivial, since $P(x > \epsilon) = 1-\epsilon$. | |
| Aug 23, 2021 at 13:51 | review | Close votes | |||
| Aug 25, 2021 at 17:32 | |||||
| Aug 23, 2021 at 13:49 | comment | added | TheGrandDuke | Can you give me a reference to this fact? | |
| Aug 23, 2021 at 13:47 | comment | added | Random | I don't think this really appropriate for mathoverflow, but the answer is yes: for each $\frac{1}{2} > \varepsilon > 0$ we have $\mathbb{P} \left( \{ a_n \} > \varepsilon \right) \to 1 - 2 \varepsilon$. Taking $\varepsilon$ arbitrarily small we get the desired result. | |
| Aug 23, 2021 at 13:38 | comment | added | TheGrandDuke | My intuition says yes, but I don't want to reinvent the wheel, if someone already did it. | |
| Aug 23, 2021 at 13:34 | history | edited | TheGrandDuke | CC BY-SA 4.0 |
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| Aug 23, 2021 at 13:30 | comment | added | user44191 | This seems like it should be trivial from the definition of "u.d. mod 1". No? (I assume you mean something equivalent to "equidistributed sequence"). | |
| Aug 23, 2021 at 13:04 | review | First posts | |||
| Aug 23, 2021 at 13:30 | |||||
| Aug 23, 2021 at 13:03 | history | edited | TheGrandDuke | CC BY-SA 4.0 |
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| Aug 23, 2021 at 12:57 | history | asked | TheGrandDuke | CC BY-SA 4.0 |