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PhilGoetz comments on Causality does not imply correlation - Less Wrong
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Yes. But it's not deep; I recommend trying yourself before consulting the answer. It follows straightforwardly from the fact that the integral of x(dx/dt) is (x^2)/2. The rest is bookkeeping to eliminate edge cases.
I didn't trouble to state the result with complete precision in the OP. For reference, here is an exact formulation (Theorem 2 of the linked note):
Let x be a differentiable real function. If the averages of x and dx/dt over the whole real line exist, and the correlation of x and dx/dt over the whole real line exists, then the correlation is zero.
Sorry that I sounded dismissive. It's a nice proof, and it wasn't obvious to me.
I am uncomfortable with using Pearson correlation to mean correlation. Consider y=sin(x), dy/dx = cos(x). These are "uncorrelated" according to Pearson correlation, but given one, there are at most 2 possibilties for the other. So knowing one gives you almost complete info about the other. So calling them "independent" seems wrong.