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Do Iterated logarithm and Law of the iterated logarithm have something in common despite the name? -- Abdull 13:31, 3 March 2006 (UTC)
I put the late Leo Breiman's name in Wiki cites. Would someone with better acquaintance with Breiman and his works be willing to start an article about him? He was a distinguished statistician, and deserves recognition. Bill Jefferys 22:40, 26 April 2006 (UTC)
I like the idea of the picture added recently by User:Dean P Foster, but looking more closely I got puzzled. The law of the iterated logarithm states that Sn is sometimes of order (but not more). Most of the time, however, it is of order The latter is probably what we see on the picture. The former is probably what we should see on a relevant picture but do not see on this picture. The logarithmic curve should be the envelope of rare large deviations (short peaks). Boris Tsirelson ( talk) 08:43, 14 May 2009 (UTC)
I start to understand; it happens because of a very nonlinear (logarithmic, in fact) vertical axis. The point is that on such a plot the graph of is quite close to the graph of even though their ratio tends to infinity. For not puzzling the reader it is probably better to show both curves on the picture and to add some words of explanation. Boris Tsirelson ( talk) 09:00, 14 May 2009 (UTC)
The picture and its caption are still not clear. Contrary to the comments above, there does not appear to be any explanation of the axes in text. My understanding is that the x-axis is and the y-axis is . There is also some logarithmic scaling of the axes in plotting, but the exact transformation eludes me. In any case, my guess of the y-axis seems to match the way the blue lines converge to zero as n tends to infinity. Would it be better to (1) change y-axis to so that the blue lines remain parallel, and (2) state explicitly the axes definitions and plotting scaling? — Preceding unsigned comment added by 67.164.26.84 ( talk) 07:01, 1 October 2013 (UTC)
The figure might be clearer if it is scaled to the variance given by CLT. That is, show in red, contant 1 in blue, and in green. — Preceding unsigned comment added by 67.164.26.84 ( talk) 17:02, 15 October 2013 (UTC)
I find the picture, though pretty, misleading in the context of the main article. The law of the iterated logarithm is an asymptotic result, which cannot be captured in a graph. Indeed, the picture shows the sum (or is it an average here?) regularly exceeding the asymptotic bound, which it cannot do when n increases without bound. So what is the point of the picture? Again, it does look good, but it invites the reader to somehow apply the law to finite n, where it in fact does not apply. Chafe66 ( talk) 22:43, 9 December 2015 (UTC)
So, can this be used to construct the confidence intervals? For example, instead of saying that
we could have been saying that
which is a nice alternative, because frankly speaking the number “95%” is quite arbitrary… For practical sample sizes the two quantities are quite similar (which could be the reason why 0.95 was chosen in the first place), in particular they are same when n = 921. Of course this all would depend on the rate of convergence of the left-hand side to the limit — anybody knows the performance of the limiting quantity in the mid-size samples? … stpasha » 22:12, 6 December 2009 (UTC)
The article claims, that
However, the article on convergence of random variables claims, that from almost sure convergence follows convergence in probability. How is the above claim possible?
Also, the text that follows the above equality, is
and it implies that in fact
Is this true? How does it follow from the Law of iterated logarithm? —Preceding unsigned comment added by 193.77.126.73 ( talk) 16:52, 21 December 2009 (UTC)
I don't know the first thing about advanced math, but I do know this article's a little puny. Anyone with those sources should be able to build it up some. Ten Pound Hammer, his otters and a clue-bat • ( Otters want attention) 20:50, 20 January 2011 (UTC)
Could we get a reference or link for the notation used, particularly the "-> forall" bit? Intuitively I know what it means (and I kinda like it) but more info would be nice, especially for people visiting this page who might not be familiar with that symbol. 24.220.188.43 ( talk) 13:28, 15 April 2011 (UTC)
One passage reads:
"Let {Yn} be independent, identically distributed random variables with means zero and unit variances. Let Sn = Y1 + … + Yn.
. . .
. . .
There are two versions of the law of large numbers —
the weak and
the strong — and they both claim that the sums Sn, scaled by n−1, converge to zero, respectively
in probability and
almost surely:
But, these statements about the two kinds of limiting behavior of Sn/n ought to be saying that as n →∞, Sn/n → 1 (not 0).
Maybe someone knowledgeable on the subject can fix this.
Daqu (
talk) 16:34, 11 February 2013 (UTC)
At the end of the Discussion section, this statement appears:
"Thus, although the quantity is less than any predefined ε > 0 with probability approaching one, that quantity will nevertheless be dropping out of that interval infinitely often, and in fact will be visiting the neighborhoods of any point in the interval (0,√2) almost surely."
1) Clearly the first part of this sentence (before the comma) cannot be true, since for any positive integer n ≥ 2, the quantity in question is positive. It appears that some condition on the index n is omitted here.
2) The part after the comma makes no sense at all, because "that quantity will nevertheless be dropping out of that interval " is meaningless gobbledygook. I have no idea which interval is meant here, and dropping out of an interval makes no mathematical sense whatsoever.
3) Finally, will be dropping out uses the future progressive tense -- a completely inappropriate tense for a mathematical statement.
Maybe someone who is knowledgeable on the subject and who can explain things clearly can fix this. Daqu ( talk) 17:01, 11 February 2013 (UTC)
In the text of the article, there's a claim that given that limsup and liminf equal to \infty and -\infty a.s., the sequence of R.V.'s can't converge in probability or a.s.:
...\limsup_n \frac{S_n}{\sqrt{n}}=\infty with probability 1. An identical argument shows that \liminf_n \frac{S_n}{\sqrt{n}}=-\infty with probability 1 as well. This implies that these quantities converge neither in probability nor almost surely:
\frac{S_n}{\sqrt n} \ \stackrel{p}{\nrightarrow}\ \forall, \qquad \frac{S_n}{\sqrt n} \ \stackrel{a.s.}{\nrightarrow}\ \forall, \qquad \text{as}\ \ n\to\infty.
I can see how the argument for no convergence a.s. goes. However, I disagree that these two facts about limsup/liminf imply no convergence in probability. Consider a sequence of R.V.'s Z_n such that Z_n = 0 with probability 1-\frac{1}{n} and Z_n = n*(-1)^n with probability \frac{1}{n}. The liminf and limsup are -\infty and +\infty with probability 1, yet Z_n \to 0 in probability.
Note: I do not claim that there exists a RV such that the sequence in the article converges to it in probability, simply that there seems to be a very large leap in reasoning here that is easy to provide a counterexample to. Can someone clarify the argument or provide a reference specific to this part of the article?
P.S. Apologies for the formatting - I can't figure out how to make it work like TEX.
165.124.129.146 ( talk) 19:39, 26 January 2015 (UTC)Sergey
The caption says the n^(-1/2) variance is given by the CLT. but this is the exact variance. also the caption refers to bounds given by the LLN, but I usually think of the CLT as giving this scaling factor. — Preceding unsigned comment added by Snarfblaat ( talk • contribs) 20:05, 4 May 2015 (UTC)
Kolmogorov's zero–one law is invoked to assert that for any fixed M, the probability that the event occurs is 0 or 1. However, the sequence is not a sequence of i.i.d. random variables, so the hypotheses of Kolmogorov's zero–one law are not satisfied (or if one wants to consider the sequence , then the considered event is not a tail event). I think one should rather invoke Hewitt–Savage zero–one law. — Preceding unsigned comment added by Lderafe ( talk • contribs) 23:02, 13 October 2016 (UTC)
Is there any good reason to use in this article the symbol (which is really a binary function , but [mis]used by different people to denote , or , depending on their preferences) instead of the unambiguous ? — Mikhail Ryazanov ( talk) 03:10, 27 December 2017 (UTC)
The last graph ( |Exhibition of Limit Theorems and their interrelationship) seems to have switched the description of the LLN and CLT - the upper-right picture seems to be the graph for CLT (description gives it as LLN), while the lower-right picture is the graph for LLN (convergence to a constant; description gives it as CLT). Oragonof ( talk) 11:07, 17 February 2021 (UTC)
I am not an expert, but my understanding is that LIL does not always hold under finite mean and variances. Doesn't there have to be some conditions on other moments? Or alternatively, boundedness of the random variables? — Preceding unsigned comment added by Cihan ( talk • contribs) 22:27, 13 March 2022 (UTC)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||
|
Do Iterated logarithm and Law of the iterated logarithm have something in common despite the name? -- Abdull 13:31, 3 March 2006 (UTC)
I put the late Leo Breiman's name in Wiki cites. Would someone with better acquaintance with Breiman and his works be willing to start an article about him? He was a distinguished statistician, and deserves recognition. Bill Jefferys 22:40, 26 April 2006 (UTC)
I like the idea of the picture added recently by User:Dean P Foster, but looking more closely I got puzzled. The law of the iterated logarithm states that Sn is sometimes of order (but not more). Most of the time, however, it is of order The latter is probably what we see on the picture. The former is probably what we should see on a relevant picture but do not see on this picture. The logarithmic curve should be the envelope of rare large deviations (short peaks). Boris Tsirelson ( talk) 08:43, 14 May 2009 (UTC)
I start to understand; it happens because of a very nonlinear (logarithmic, in fact) vertical axis. The point is that on such a plot the graph of is quite close to the graph of even though their ratio tends to infinity. For not puzzling the reader it is probably better to show both curves on the picture and to add some words of explanation. Boris Tsirelson ( talk) 09:00, 14 May 2009 (UTC)
The picture and its caption are still not clear. Contrary to the comments above, there does not appear to be any explanation of the axes in text. My understanding is that the x-axis is and the y-axis is . There is also some logarithmic scaling of the axes in plotting, but the exact transformation eludes me. In any case, my guess of the y-axis seems to match the way the blue lines converge to zero as n tends to infinity. Would it be better to (1) change y-axis to so that the blue lines remain parallel, and (2) state explicitly the axes definitions and plotting scaling? — Preceding unsigned comment added by 67.164.26.84 ( talk) 07:01, 1 October 2013 (UTC)
The figure might be clearer if it is scaled to the variance given by CLT. That is, show in red, contant 1 in blue, and in green. — Preceding unsigned comment added by 67.164.26.84 ( talk) 17:02, 15 October 2013 (UTC)
I find the picture, though pretty, misleading in the context of the main article. The law of the iterated logarithm is an asymptotic result, which cannot be captured in a graph. Indeed, the picture shows the sum (or is it an average here?) regularly exceeding the asymptotic bound, which it cannot do when n increases without bound. So what is the point of the picture? Again, it does look good, but it invites the reader to somehow apply the law to finite n, where it in fact does not apply. Chafe66 ( talk) 22:43, 9 December 2015 (UTC)
So, can this be used to construct the confidence intervals? For example, instead of saying that
we could have been saying that
which is a nice alternative, because frankly speaking the number “95%” is quite arbitrary… For practical sample sizes the two quantities are quite similar (which could be the reason why 0.95 was chosen in the first place), in particular they are same when n = 921. Of course this all would depend on the rate of convergence of the left-hand side to the limit — anybody knows the performance of the limiting quantity in the mid-size samples? … stpasha » 22:12, 6 December 2009 (UTC)
The article claims, that
However, the article on convergence of random variables claims, that from almost sure convergence follows convergence in probability. How is the above claim possible?
Also, the text that follows the above equality, is
and it implies that in fact
Is this true? How does it follow from the Law of iterated logarithm? —Preceding unsigned comment added by 193.77.126.73 ( talk) 16:52, 21 December 2009 (UTC)
I don't know the first thing about advanced math, but I do know this article's a little puny. Anyone with those sources should be able to build it up some. Ten Pound Hammer, his otters and a clue-bat • ( Otters want attention) 20:50, 20 January 2011 (UTC)
Could we get a reference or link for the notation used, particularly the "-> forall" bit? Intuitively I know what it means (and I kinda like it) but more info would be nice, especially for people visiting this page who might not be familiar with that symbol. 24.220.188.43 ( talk) 13:28, 15 April 2011 (UTC)
One passage reads:
"Let {Yn} be independent, identically distributed random variables with means zero and unit variances. Let Sn = Y1 + … + Yn.
. . .
. . .
There are two versions of the law of large numbers —
the weak and
the strong — and they both claim that the sums Sn, scaled by n−1, converge to zero, respectively
in probability and
almost surely:
But, these statements about the two kinds of limiting behavior of Sn/n ought to be saying that as n →∞, Sn/n → 1 (not 0).
Maybe someone knowledgeable on the subject can fix this.
Daqu (
talk) 16:34, 11 February 2013 (UTC)
At the end of the Discussion section, this statement appears:
"Thus, although the quantity is less than any predefined ε > 0 with probability approaching one, that quantity will nevertheless be dropping out of that interval infinitely often, and in fact will be visiting the neighborhoods of any point in the interval (0,√2) almost surely."
1) Clearly the first part of this sentence (before the comma) cannot be true, since for any positive integer n ≥ 2, the quantity in question is positive. It appears that some condition on the index n is omitted here.
2) The part after the comma makes no sense at all, because "that quantity will nevertheless be dropping out of that interval " is meaningless gobbledygook. I have no idea which interval is meant here, and dropping out of an interval makes no mathematical sense whatsoever.
3) Finally, will be dropping out uses the future progressive tense -- a completely inappropriate tense for a mathematical statement.
Maybe someone who is knowledgeable on the subject and who can explain things clearly can fix this. Daqu ( talk) 17:01, 11 February 2013 (UTC)
In the text of the article, there's a claim that given that limsup and liminf equal to \infty and -\infty a.s., the sequence of R.V.'s can't converge in probability or a.s.:
...\limsup_n \frac{S_n}{\sqrt{n}}=\infty with probability 1. An identical argument shows that \liminf_n \frac{S_n}{\sqrt{n}}=-\infty with probability 1 as well. This implies that these quantities converge neither in probability nor almost surely:
\frac{S_n}{\sqrt n} \ \stackrel{p}{\nrightarrow}\ \forall, \qquad \frac{S_n}{\sqrt n} \ \stackrel{a.s.}{\nrightarrow}\ \forall, \qquad \text{as}\ \ n\to\infty.
I can see how the argument for no convergence a.s. goes. However, I disagree that these two facts about limsup/liminf imply no convergence in probability. Consider a sequence of R.V.'s Z_n such that Z_n = 0 with probability 1-\frac{1}{n} and Z_n = n*(-1)^n with probability \frac{1}{n}. The liminf and limsup are -\infty and +\infty with probability 1, yet Z_n \to 0 in probability.
Note: I do not claim that there exists a RV such that the sequence in the article converges to it in probability, simply that there seems to be a very large leap in reasoning here that is easy to provide a counterexample to. Can someone clarify the argument or provide a reference specific to this part of the article?
P.S. Apologies for the formatting - I can't figure out how to make it work like TEX.
165.124.129.146 ( talk) 19:39, 26 January 2015 (UTC)Sergey
The caption says the n^(-1/2) variance is given by the CLT. but this is the exact variance. also the caption refers to bounds given by the LLN, but I usually think of the CLT as giving this scaling factor. — Preceding unsigned comment added by Snarfblaat ( talk • contribs) 20:05, 4 May 2015 (UTC)
Kolmogorov's zero–one law is invoked to assert that for any fixed M, the probability that the event occurs is 0 or 1. However, the sequence is not a sequence of i.i.d. random variables, so the hypotheses of Kolmogorov's zero–one law are not satisfied (or if one wants to consider the sequence , then the considered event is not a tail event). I think one should rather invoke Hewitt–Savage zero–one law. — Preceding unsigned comment added by Lderafe ( talk • contribs) 23:02, 13 October 2016 (UTC)
Is there any good reason to use in this article the symbol (which is really a binary function , but [mis]used by different people to denote , or , depending on their preferences) instead of the unambiguous ? — Mikhail Ryazanov ( talk) 03:10, 27 December 2017 (UTC)
The last graph ( |Exhibition of Limit Theorems and their interrelationship) seems to have switched the description of the LLN and CLT - the upper-right picture seems to be the graph for CLT (description gives it as LLN), while the lower-right picture is the graph for LLN (convergence to a constant; description gives it as CLT). Oragonof ( talk) 11:07, 17 February 2021 (UTC)
I am not an expert, but my understanding is that LIL does not always hold under finite mean and variances. Doesn't there have to be some conditions on other moments? Or alternatively, boundedness of the random variables? — Preceding unsigned comment added by Cihan ( talk • contribs) 22:27, 13 March 2022 (UTC)