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Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^m$ be two independent random vectors. Then, say that we have a third real valued random variable $Z=f(X,Y)$, with $f$ being measurable.

  • Say that we know that $Z\perp X$. Can we then say that $f(X,Y)$ is constant in $X$ in some sense?
  • Alternatively, say that for $Y\in S$ ($S$ with positive probability), $f(X,Y)$ is not constant in $X$. Is it true then that $f(X,Y)$ is not independent of $X$?

I've seen a similar question in this link: answers in this link show that if $X$ is independent of $f(X)$, then $f(X)$ must be a constant almost surely. Yet, I found it hard to adapt this argument to this case of a function $f(X,Y)$.

random_name
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1 Answers1

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No, independence of $X$ and $Z$ doesn't imply independence conditional on $Y$ (equivalent to $f(X,Y)$ being constant in $X$).

For example, suppose that $X$ and $Y$ are independent and both take values of $-1$ and $1$ with equal probability. Suppose that $Z=f(X,Y)=XY$. You then have the following four outcomes occuring with equal probabilities $$ \begin{matrix} X & Y & Z \\ -1 & -1 & 1 \\ -1 & 1 & -1 \\ 1 & -1 & -1 \\ 1 & 1 & 1 \end{matrix} $$ Although $X$ and $Z=f(X,Y)$ are clearly independent, $f(X,Y)$ is not constant in $X$.

The second statement is the contraposition of the first statement and so logically equivalent and not true.

Jarle Tufto
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