I asked this question on MO, but no answer.

I don't know if this definition has been already given.

Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. We say $Y$ is strongly dependent on $X$ if for any real-valued non-degenerate function acting on $\mathcal{Y}$, $f(Y)$ is also dependent on $X$.

Here I need to assume that $f(Y)$ is non-degenerate because if $f$ is a constant function then $f(Y)$ in independent of both $X$ and $Y$.

What can we know about $P_{XY}$ if $Y$ is strongly dependent on $X$?

Any necessary and sufficient condition for $Y$ being strongly dependent on $X$?

  • 4
    $\begingroup$ By "dependent" do you just mean "not independent"? $\endgroup$ – Sasho Nikolov Sep 28 '14 at 3:19
  • $\begingroup$ Link to MO question? $\endgroup$ – Tyson Williams Sep 28 '14 at 12:25
  • $\begingroup$ @SashoNikolov, yes by "dependent" I mean "not independent". $\endgroup$ – SAmath Oct 8 '14 at 14:29

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