I tried, but it didn’t work for me. I could make a codecogs URL to exhibit the image in my browser, but it got munged when I tried the  embedding.
The problem must be in escape character (see the last section of the wiki article). Try copy-pasting the code I gave above in your comment, and notice the placement of backslashes.
where N is normalisation; it seems that you suppose that =0 and <f’> finite, then. =0 follows from boundedness, but for the derivative it’s not clear. If <f’> on (a,b) grows more rapidly than (b-a), anything can happen.
This cannot happen. f is assumed bounded. Therefore the average of f’ over the interval [a,b] tends to zero as the bounds go to infinity.
The precise, complete mathematical statement and proof of the theorem does involve some subtlety of argument (consider what happens if f = sin(exp(x))) but the theorem is correct.
Theorem: In the long run, a bounded, differentiable real function has zero correlation with its first derivative.
I don’t understand the theorem. What does “in the long run” mean? Is it that in limit a,b->\infty
(\int{a,b} f(x)f’(x) dx)/(b-a)=(\int{a,b} f(x) dx)(\int_a^b f’(y) dy)/(b-a)^2 ?
Sorry for the quasi-TEX notation, even the underscore doesn’t appear here. Is there any elegant way to write formulae on LW?
Not quite, it’s that as a and b go to infinity,
(\int_{a,b}f(x)f’(x)dx)/(b-a))
goes to zero. \int_{a,b}f(x)f’(x)dx = [ f(x)^2/2 ]^b_a, which is bounded, while b-a is unbounded, QED.
LaTeX to Wiki might work, but LaTeX to LW comment doesn’t.
Source code:
Formatting tutorial on the Wiki
I tried, but it didn’t work for me. I could make a codecogs URL to exhibit the image in my browser, but it got munged when I tried the  embedding.
The problem must be in escape character (see the last section of the wiki article). Try copy-pasting the code I gave above in your comment, and notice the placement of backslashes.
The standard form for correlation coefficient is
cov(x,y)=N(-)
where N is normalisation; it seems that you suppose that =0 and <f’> finite, then. =0 follows from boundedness, but for the derivative it’s not clear. If <f’> on (a,b) grows more rapidly than (b-a), anything can happen.
This cannot happen. f is assumed bounded. Therefore the average of f’ over the interval [a,b] tends to zero as the bounds go to infinity.
The precise, complete mathematical statement and proof of the theorem does involve some subtlety of argument (consider what happens if f = sin(exp(x))) but the theorem is correct.
See the description on the Wiki of how to include LaTeX in comments.