Comments on: The strong law of large numbers http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Fri, 24 May 2013 18:38:29 +0000 hourly 1 http://wordpress.com/ By: Mate Wierdl http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-228297 Wed, 08 May 2013 13:59:17 +0000 http://terrytao.wordpress.com/?p=416#comment-228297 Hi Terry! Maybe a few exercises could be added based on the following remarks:

If we assume finite second moment then only “multiplicativity” is needed. By multiplicativity, I mean \mathbb EX_iX_j=\mathbb E X_i \mathbb E X_j. This translates to orthogonality if the expectations are $\latex 0$.

An application would be Weyl’s result: if n_1<n_2<n_3<\dots is a sequence of integers then for almost all \alpha, the sequence n_1\alpha, n_2\alpha, n_3\alpha,dots is uniformly distributed \mod 1. For this, we take X_i=\exp(2\pi i n_i \alpha).

With this mutiplicativity assumption, one can simply get results for random variables with varying expectations: Consider nonnegative random variables with varying expectation, but then we would divide by the expectation of the sum of the first $n$ random variables instead of n to get the right normalization. For example, the expectations can go to 0 arbitrary slowly as long as \sum_i \mathbb EX_i=\infty. But the expectations can also go to \infty! They cannot increase arbitrary fast: something like \mathbb EX_i = O(2^{i^b}) for b<1/2 seems to be the limit for a strong law, while for norm (weak) convergence, one can get arbitrary close to 2^i, that is we can have \mathbb E X_i = 2^{o(i)}.

I think these are good exercises for using and testing the limits of this "lacunary subsequence" trick.

]]> By: Laws of large numbers and Birkhoff’s ergodic theorem | Vaughn Climenhaga's Math Blog http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-219053 Sun, 10 Mar 2013 03:36:12 +0000 http://terrytao.wordpress.com/?p=416#comment-219053 [...] including a different proof of the weak law than the one above, can be found on Terry Tao’s blog), we observe that the strong law of large numbers can be viewed as a special case of the Birkhoff [...]

]]> By: Iosif Pinelis http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-215916 Wed, 06 Feb 2013 04:01:15 +0000 http://terrytao.wordpress.com/?p=416#comment-215916 Thank you for the help with LaTeX. A couple more points here:

(i) Of course, I wanted to say “the pairwise independence does not imply the stationarity, even if the X_i‘s are identically distributed”, rather than just “the pairwise independence does not imply the stationarity”.

(ii) Tao’s result, with the pairwise independence rather than with the complete independence, can be extended in a standard manner to the case when the X_i‘s take values in an arbitrary separable Banach space (say).

]]> By: Iosif Pinelis http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-215780 Tue, 05 Feb 2013 05:05:44 +0000 http://terrytao.wordpress.com/?p=416#comment-215780 I think this result and proof are very nice. There is a vague commonality between Keane’s and Tao’s proof: both exploit, to an extent, the fact that the sample mean \bar{X}_n varies little, in a sense, with n. It is also nice that Tao’s result is not contained in the ergodic theorem. Indeed, one can easily see that the pairwise independence does not imply the stationarity. E.g., let X_1,\dots,X_6 be defined as R_1, R_2, R_1R_2, R_1R_3, R_2R_3, R_1R_2R_3, respectively, where the R_i‘s are independent Rademacher random variables, each taking each of the values 1, -1 with probability 1/2. Define X_7,\dots,X_{12} similarly, based on R_4, R_5, R_6; etc. Then the X_i‘s are pairwise independent. However, P(X_1X_2X_3=1)=1, whereas P(X_4X_5X_6=1)=1/2. So, the sequence (X_i) is not stationary.

[LaTeX code corrected; the problem was the lack of a space between "latex" and the LaTeX code. Also, the curly braces were unnecessary. -T]

]]> By: abc http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-123960 Sat, 21 Jan 2012 23:00:33 +0000 http://terrytao.wordpress.com/?p=416#comment-123960 It seems like the proof of the pointwise ergodic theorem is easier than this proof! Inspecting that proof, it uses the the added generality of measure-preserving systems to deal with functions that you could not talk about (naturally) in the setting of the strong law of large numbers.

]]> By: Note on the weak and strong laws of large numbers | Mathematics and me-themed antics. http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-101470 Mon, 07 Nov 2011 14:02:55 +0000 http://terrytao.wordpress.com/?p=416#comment-101470 [...] my development was different from that in the book. My notes essentially follow the proof given by Terry Tao, so you may want to refer to his notes as well. Like this:LikeBe the first to like this post. [...]

]]> By: The law of large numbers and the central limit theorem | Nair Research Notes http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-70957 Sun, 28 Aug 2011 14:06:28 +0000 http://terrytao.wordpress.com/?p=416#comment-70957 [...] central limit theorem (CLT). There are excellent resources on the net for LLN and CLT. For example, this and this are highly recommended readings. This blog will play a complementary with figures and [...]

]]> By: lkozma http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-70948 Sun, 28 Aug 2011 12:34:14 +0000 http://terrytao.wordpress.com/?p=416#comment-70948 Thank you for the clear and informative post!

The formula for linearity of expectation shouldn’t have parentheses on the left side?

]]> By: Russell Lyons http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-46554 Thu, 19 Aug 2010 15:09:30 +0000 http://terrytao.wordpress.com/?p=416#comment-46554 Hi, Terry.

I just happened across this post. I have two small comments:

1. For the proof of the Borel-Cantelli lemma, one can simply observe that since the expectation of the sum of indicators is finite, so is the sum itself a.s. For some reason, it has been more popular to do it your way (i.e., using Markov’s inequality).

2. For those who are interested in refinements of the interpolation method that apply to weakly dependent random variables under a second moment condition, I can suggest an old paper of mine: Strong laws of large numbers for weakly correlated random variables, Mich. Math. J. 35, No. 3 (1988), 353–359 (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.mmj/1029003816).

Best,
Russ

]]> By: The Khinchine Law of Large Numbers | The Longboat and the Otter http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/#comment-44871 Sat, 15 May 2010 20:37:05 +0000 http://terrytao.wordpress.com/?p=416#comment-44871 [...] http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ [...]

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