Timeline for "Ergodicity" of non-invariant product measures (with respect to the shift)

7 events

when toggle format	what		by	license	comment
May 9, 2017 at 1:48	comment	added	Anthony Quas		Sorry. I can't because the counter-example was created specifically in response to this question. But: if you agree with the calculation that $f(T^n(x_0,x_1,\ldots))=\sin^2(2\pi x_n)\sin(2\pi x_{2^j})$ for all realizations $(x_0,x_1,\ldots)$ of the random vector $(X_0,X_1,\ldots)$, then the reason for the lack of convergence is that the average over the block $[2^{j-1},2^j)$ is close to $\frac 12\sin(2\pi x_{2^j})$; and as these blocks are doubling in length, brings the overall average towards that value; the next block brings the overall average towards $\frac 12\sin(2\pi x_{2^{j+1}})$ etc.
May 8, 2017 at 23:46	comment	added	user127022		could you perhaps point me to a reference where this (counter-)example appears in more detail?
Apr 28, 2017 at 3:43	history	edited	Anthony Quas	CC BY-SA 3.0	edited body
Apr 28, 2017 at 3:42	comment	added	Anthony Quas		Yes, absolutely. Thanks for the correction.
Apr 27, 2017 at 16:56	comment	added	user127022		Should it be $f(T^n(x_0,x_1,\dots)) = \sin^2(2\pi x_n)\sin(2\pi x_{2^j - \lfloor x_n\rfloor})$ for $x_n \in [2^{j-1}, 2^j)$?
Apr 24, 2017 at 5:51	history	edited	Anthony Quas	CC BY-SA 3.0	added 102 characters in body
Apr 23, 2017 at 22:15	history	answered	Anthony Quas	CC BY-SA 3.0