When can a function of random variables be independent of the random variables?

Question

Let $Z\equiv (z_{1}, \ldots, z_{T})$ be a vector of random variables. Is it possible that $f(Z)$ be independent of $Z$ when $f(Z)$ is not a constant. Also, if the above is true, under what general condition on $Z$ is $f(Z)\perp Z$.

Answer 1

No.

If $f(Z)$ is not a constant then we can find a Borel-set $B$ with $P(f(Z)\in B)\in(0,1)$.

Then also $P(Z\in f^{-1}(B^{\complement}))=P(f(Z)\in B^{\complement})\in(0,1)$.

Then $P(f(Z)\in B\wedge Z\in f^{-1}(B^{\complement}))=P(\varnothing)=0$.

But independence implies that: $$P(f(Z)\in B\wedge Z\in f^{-1}(B^{\complement}))=P(f(Z)\in B)P(Z\in f^{-1}(B^{\complement}))$$ which cannot be true because the LHS takes value $0$ and the RHS is positive.