Correlation of $X^3$ and $Y$

Question

Suppose that $X$ and $Y$ are two uncorrelated random variables and the PDF of them are both even. Can we expect that $X^3$ and $Y$ are also uncorrelated? Thanks.

Answer 1

No. We definitely can not expect that.

Consider $X$ and $Y$ has the following pmf on four points $\{ (-1,-1), (2,-1), (1,1),(-2,1)\}$: $P_{(-2,1)}=P_{(2,-1)}=1/6$ and $P_{(1,1)}=P_{(-1,-1)}=1/3$.

Then clearly we have $X,Y$ are uncorrelated, with zero mean and even marginal distributions. And what's more, this distribution is symmetric with respect to the origin. But $X^3,Y$ are correlated:

$$E[X^3 Y] = 2 \ne E[X^3] E[Y] =0$$

If worried about this is PMF not PDF, you could substitute each point into a uniform square of radius $0.1$.

Answer 2

We cannot expect $X^3$ and $Y$ uncorrelated.

Like here, choose $X\sim N(0,1)$ (standard normal), and

$$ Y= \begin{cases} X & \text{ if } |X|\le c\\ -X & \text{ if } |X|> c\\ \end{cases} $$ where $c\approx1.54$ is chosen such that $E[XY]=0$

Then it is easy to prove that $Y\sim N(0,1)$, so both PDFs are even, $X$ and $Y$ are uncorrelated, and furthermore:

$E[X^3Y]\approx -2.2\ne E[X^3]E[Y]=0$, so $X^3$ and $Y$ are correlated.