<p>I asked <a href="https://mathoverflow.net/questions/181840/strong-dependence">this question on MO</a>, but no answer.</p>

<p>I don't know if this definition has been already given. </p>

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

<p>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 $Y$. </p>

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

<p>Any necessary and sufficient condition for $Y$ being strongly dependent on $X$?    </p>

CC BY-SA 3.0
2014-09-28T03:19:21.700Z
By "dependent" do you just mean "not independent"?
Sasho Nikolov
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2014-09-28T12:25:34.307Z
Link to MO question?
Tyson Williams
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2014-10-08T14:29:30.267Z
@SashoNikolov, yes by "dependent" I mean "not independent".
SAmath
<reference-request><pr.probability><it.information-theory><st.statistics>
Strong Dependence
SAmath
# Strong Dependence

(Asked by: username_0 on 2014-09-27)

I asked [this question on MO](https://mathoverflow.net/questions/181840/strong-dependence), 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 $X$?

**Tags:** reference-request, pr.probability, it.information-theory, st.statistics

> **Comments:**
> **username_1**: By "dependent" do you just mean "not independent"?  
> **username_2**: Link to MO question?  
> **username_0**: @SashoNikolov, yes by "dependent" I mean "not independent".  

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## 0 answers

