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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 $X$?

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    $\begingroup$ The problem is that the constant function $f(Y) = 4$ is always independent of $X$, so nothing can satisfy this definition (right?). $\endgroup$
    – usul
    Commented Sep 26, 2014 at 4:12
  • $\begingroup$ @usul I guess to make it interesting we can add the condition that $ f (Y) $ be dependent on $ Y $. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Sep 26, 2014 at 7:12
  • $\begingroup$ I meant any function such that $f(Y)$ is non-degenerate so in other words, $f(Y)$ take at least two different values with no-zero probabilities. Then this $f(Y)$ will depend on $Y$ as @BjørnKjos-Hanssen mentioned. $\endgroup$
    – math-Student
    Commented Sep 26, 2014 at 14:34
  • $\begingroup$ What sort of answer are you looking for? It suffices to consider functions $f$ taking precisely two values. From this you can get that a necessary and sufficient condition is that for all $A$ with $\emptyset\subset A\subset \mathcal{Y}$, there exists $x$ such that $p_{XY}(\{x\}\times A) \ne p_X(\{x\})p_Y(A)$. $\endgroup$
    – James Martin
    Commented Sep 29, 2014 at 13:43
  • $\begingroup$ First I was wondering if any references looked into this definition. The name "strong dependence" has been used by different properties. I am particularly interested in information theory point of view. $\endgroup$
    – math-Student
    Commented Sep 29, 2014 at 20:38

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