Talk:Law of the iterated logarithm

message: Do Iterated logarithm and Law of the iterated logarithm have something in common despite the name? --Abdull 13:31, 3 March 2006 (UTC)
replies: text: No, nothing in common, I believe. Boris Tsirelson (talk) 08:50, 14 May 2009 (UTC)
thread_title:  Iterated logarithm? 

message: 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)
thread_title:  Leo Breiman 

message: So, can this be used to construct the confidence intervals? For example, instead of saying that
: 
    \frac{1}{\sqrt{n}} |S_n| < 1.96 \qquad \text{with probability }95\%,
  
we could have been saying that
: 
    \frac{1}{\sqrt n} |S_n| < \sqrt{2\log\log n} \qquad \text{almost surely},
  
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)
replies: text: I am afraid, it could be drastically different from the whole philosophy of statistics; definitely, original research (to be discussed elsewhere, if you like). "Arbitrary" numbers like 0.95 are inevitable (according to that philosophy) and have important practical meaning: the probability of rejecting a true hypothesis. Boris Tsirelson (talk) 20:30, 22 December 2009 (UTC)
thread_title:  practical use? 

message: 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)
thread_title:  Expansion 

message: 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)
replies: text: I repeat (see several lines above): "I never saw such notation  {\nrightarrow}\ \forall;  is it a good idea?" I suspect it is someone's neologism (neo-notation :-) ). --Boris Tsirelson (talk) 15:08, 15 April 2011 (UTC)
thread_title:  Notation 

message: 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:
: 
    \frac{S_n}{n} \ \xrightarrow{p}\ 0, \qquad
    \frac{S_n}{n} \ \xrightarrow{a.s.} 0, \qquad \text{as}\ \ n\to\infty.
  "
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)
replies: text: Sorry, I did not understand why do you believe they tend to 1. They tend to the expectation of Y_1, assumed to be 0. Boris Tsirelson (talk) 17:10, 11 February 2013 (UTC)
replies: text: Of course you are right. The reason I thought they tend to 1 was because my brain wasn't working right.  Of course they should tend to the expectation of any of the i.i.d. Yi's.   My apologies for my stupid error.Daqu (talk) 17:51, 11 February 2013 (UTC)
thread_title:  Error in statements about Laws of Large Numbers 

message: 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)
replies: text: Oh yes, you are right, thank you. Now fixed. Boris Tsirelson (talk) 19:59, 26 January 2015 (UTC)
thread_title:  limsup=\infty implies no convergence in probability? 

message: 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)
replies: text: Yes, you are right, thank you. Now fixed. In addition: that is (plus-minus) root of variance. Boris Tsirelson (talk) 05:32, 5 May 2015 (UTC)
thread_title:  figure caption 

message: Kolmogorov's zero–one law is invoked to assert that for any fixed M, the probability that the event  \limsup_n \frac{S_n}{\sqrt{n}} > M  occurs is 0 or 1. However, the sequence  (S_n)_{n\ge 0}  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  (Y_n)_{n\ge 0} , 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)
thread_title:  Kolmogorov's zero–one law 

message: Is there any good reason to use in this article the symbol \log (which is really a binary function \log_x y, but [mis]used by different people to denote \log_2, \log_e or \log_{10}, depending on their preferences) instead of the unambiguous \ln? — Mikhail Ryazanov (talk) 03:10, 27 December 2017 (UTC)
replies: text: "log" seems to be essentially standard in the literature. IAmAnEditor (talk) 14:10, 1 January 2018 (UTC)
replies: text: Yes; and "ln" is more usual in Russia. Boris Tsirelson (talk) 15:46, 1 January 2018 (UTC)
replies: text: Nevertheless, the article Natural logarithm uses \ln throughout. And regarding being "standard in the literature", \log in computer science usually means \log_2. — Mikhail Ryazanov (talk) 20:47, 1 January 2018 (UTC)
replies: text: The literature about the Law of the iterated logarithm seems to generally use "log". For examples, see the Encyclopedia of Statistical Sciences and  the International Encyclopedia of Statistical Science, as well as books such as Limit Theorems of Probability Theory by Petrov. Wikipedia should follow the literature on this. IAmAnEditor (talk) 21:14, 1 January 2018 (UTC)
thread_title:  Why "log" instead of "ln"? 

message: 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)
thread_title:  Wrong intra-picture description of last graph 

message: 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)
replies: text: I'm looking at the Breiman cited source right now, and his Theorem 3.52 agrees with the statement in the article, as far as I can tell.  Theorem 3.52 actually has a sigma in the denominator, so variance 1 is not assumed.  It is not clear to me that Breiman even assumes mean zero, but it must be in there somewhere. Doctormatt (talk) 03:01, 14 March 2022 (UTC)
thread_title:  Is the Statement Correct?