Comments on: 254A, Lecture 8: The mean ergodic theorem http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Fri, 24 May 2013 15:11:30 +0000 hourly 1 http://wordpress.com/ By: Multiple recurrence and convergence results associated to $F_{p}^{omega}$-actions | What's new http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-230671 Wed, 22 May 2013 00:30:28 +0000 http://terrytao.wordpress.com/?p=250#comment-230671 […] e.g. this previous blog post. Informally, one can interpret this limit formula as an equidistribution result: if is drawn at […]

]]> By: Alok Bakshi http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-215615 Sun, 03 Feb 2013 18:02:14 +0000 http://terrytao.wordpress.com/?p=250#comment-215615 Dear Professor Tao, have you applied Jensen’s inequality with \phi(x) = x^2 to obtain (1), because I couldn’t see how Cauchy Schwartz inequality be applied here.

[ To show \int_X f^2\ d\mu \geq (\int_X f\ d\mu)^2, one can either use Jensen, or else apply Cauchy-Schwarz to the functions f and 1. -T]

]]> By: Terence Tao http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-197059 Wed, 28 Nov 2012 00:52:19 +0000 http://terrytao.wordpress.com/?p=250#comment-197059 Yes; I recommend the exercise of verifying that the definition given in the Wikipedia article is indeed the orthogonal projection to the Hilbert space L^2({\mathcal B}) (this is also alluded to in Section 5 of the Wikipedia article, as well as in equation (14) of the present article), and that in the case when the factor {\mathcal B} is generated by a discrete random variable, the definition collapses (outside of an event of probability zero, at least) to the definition given in Notes 0 (or at the beginning of the Wikipedia article); this is alluded to in Remark 6 of these notes.

]]> By: Jack http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-197052 Wed, 28 Nov 2012 00:12:52 +0000 http://terrytao.wordpress.com/?p=250#comment-197052 In this note, the “conditional expectation” is defined as the projection on the closed subspace of a Hilbert space. I learned another version on [this Wikipedia article] (http://en.wikipedia.org/wiki/Conditional_expectation#Formal_definition). And I also see one in 254A Note 0
(http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/#as-rem), which is for the discrete random variables. Are they essentially the same?

]]> By: Advanced Analyis, Notes 5: Hilbert spaces (Application: Von Neumann’s mean ergodic theorem) « Noncommutative Analysis http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-187542 Tue, 30 Oct 2012 08:26:28 +0000 http://terrytao.wordpress.com/?p=250#comment-187542 [...] operators-on-Hilbert-space theory, by proving von Neumann’s mean ergodic theorem. See also this treatment by Terry Tao on his [...]

]]> By: Walsh’s ergodic theorem, metastability, and external Cauchy convergence « What’s new http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-185893 Thu, 25 Oct 2012 18:10:50 +0000 http://terrytao.wordpress.com/?p=250#comment-185893 [...] averages, namely the orthogonal projection of to the -invariant factors. (See for instance my lecture notes on this theorem.) While this theorem ostensibly involves measure theory, it can be abstracted to the more general [...]

]]> By: Rex http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-173991 Sun, 23 Sep 2012 03:06:12 +0000 http://terrytao.wordpress.com/?p=250#comment-173991 Typo: where you write “Now observe (using (10)) that \frac{1}{N} \frac{\lambda^{N}-\lambda}{\lambda-1} is bounded in magnitude by 1…”

I think you want:

\frac{1}{N} \frac{\lambda^{N}-1}{\lambda-1}

[Corrected, thanks - T.]

]]> By: Terence Tao http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-173924 Sat, 22 Sep 2012 23:00:51 +0000 http://terrytao.wordpress.com/?p=250#comment-173924 (a) yes; see http://terrytao.wordpress.com/2008/01/08/254a-lecture-1-overview/

(b) Not directly, though in practice one can often derive recurrence results for non-invertible ergodic systems from the invertible case by a lifting trick, see e.g. Exercise 9 of http://terrytao.wordpress.com/2008/01/15/254a-lecture-4-multiple-recurrence/ for an instance of this.

]]> By: Rex http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-173860 Sat, 22 Sep 2012 19:20:24 +0000 http://terrytao.wordpress.com/?p=250#comment-173860 Just to make sure we have the same conventions:

In the definition of measure-preserving system, when you say “T is invertible”, you mean that the map T: X \longrightarrow X is a (measurable) bijection, so that T^{-1} is defined on the level of sets?

In this case, we are not considering examples such as the doubling map on the circle? I think some other texts only require that T satisfy \mu(T^{-1} E) = \mu(E) for all measurable E \subset X.

]]> By: Rex http://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/#comment-146643 Sun, 10 Jun 2012 11:40:10 +0000 http://terrytao.wordpress.com/?p=250#comment-146643 On second thought, I suppose the norms could be different.

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