Timeline for Definition of asymptotic frequency
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2018 at 21:25 | comment | added | Iosif Pinelis | @BenoitSanchez : To an extent, this discussion is indeed necessarily informal, but this goes back to terms such as "reasonable" used in your question, and such terms seem to be integral to the question. Also, I asked what you meant by "most often"; this question seems to make sense here. | |
| Mar 29, 2018 at 20:31 | comment | added | Benoit Sanchez | I answered to the remarks that the frequency was between 1/3 and 2/3 and to the idea that the $x_n$ was symmetric in 0/1 (which are false for this example). It was just to clarify the example. Now, to the rest is willingly informal and arguable. | |
| Mar 29, 2018 at 20:20 | comment | added | Iosif Pinelis | As for the convergence of $f_n$ in probability (which concerns only the one-dimensional distributions of the random variable $f_n$) in your example, it does not seem of sufficient relevance to the behavior of the sequence $(f_n)$, which is an infinite-dimensional random vector. This should be especially true in your example, where the convergence of (say) $Ef_n$ to $0$ is very slow, of a $\log\log n$ rate. | |
| Mar 29, 2018 at 20:06 | comment | added | Iosif Pinelis | @BenoitSanchez : That was not how I interpreted your original post. You are saying "The frequency is most often very small"; in what sense most often? Every once in a while, your frequency stays at a level $\ge 1/4$ during at least $1/4$ of the entire prehistory. Whatever the frequency "loses" during the very-very long first $1/2$ of such a journey, it recovers -- gradually and steadily -- during the last very-very long $1/2$ of it. (You can't see this if you compress the blocks into single digits.) | |
| Mar 29, 2018 at 18:45 | history | edited | Benoit Sanchez |
edited tags
|
|
| Mar 29, 2018 at 18:16 | history | edited | Benoit Sanchez | CC BY-SA 3.0 |
added 150 characters in body
|
| Mar 29, 2018 at 18:13 | comment | added | Benoit Sanchez | Actually, my example is: for block #$k$ decide it is 1s with probability $1/k$, 0s otherwise (like coin tossing). There is no symmetry. The frequency is most often very small, except after seeing a group of 1s. | |
| Mar 29, 2018 at 18:02 | comment | added | Sridhar Ramesh | Right; there's such symmetry between 1s and 0s in this example, that, if anything, I'd only want to assign a generalized frequency of 50-50 between the two (as in a kind of Cesàro-style generalized limit). | |
| Mar 29, 2018 at 15:55 | comment | added | Iosif Pinelis | I don't see any example where " the limit does not exist yet it sounds "reasonable" to say that the asymptotic frequency still exists". In your diadic-blocks example, the relative frequency of 1's never goes below $1/3$, and it oscillates asymptotically, with ever doubling "periods", between $1/3$ and $2/3$. So, I cannot see why "it could make sense saying the asymptotic frequency is 0" here. | |
| Mar 29, 2018 at 15:39 | answer | added | Iosif Pinelis | timeline score: 2 | |
| Mar 28, 2018 at 14:45 | answer | added | Piotr Hajlasz | timeline score: 4 | |
| Mar 28, 2018 at 14:13 | history | asked | Benoit Sanchez | CC BY-SA 3.0 |