0
$\begingroup$

Consider a random vector $\mathbf{Z}$ of length $N$ whose elements are i.i.d. and zero-mean. We construct two new scalar random variable $X$ and $Y$ from $\mathbf{Z}$ as follows:

$X = \langle\mathbf{a}, \mathbf{Z}\rangle = \mathbf{a}^T\mathbf{Z}$ and $Y = \langle\mathbf{b}, \mathbf{Z}\rangle = \mathbf{b}^T\mathbf{Z}$ where $\langle,\rangle$ indicates the inner product, and $\mathbf{a}$ and $\mathbf{b}$ are deterministic and orthogonal vectors, i.e., $\langle\mathbf{a},\mathbf{b}\rangle=0$.

One can easily show $X$ and $Y$ are uncorrelated, my question is we can generally show that $X$ and $Y$ are also independent!!!

I mean that for the case that $\mathbf{Z}$ follows jointly Gaussian distribution, $X$ and $Y$ are independent but what about other distributions!!!

Best, Farzad

$\endgroup$
5
$\begingroup$

OK, I respond because I need some reputation points...

Counterexample: $N = 2$, $a = (1,1)$, $b = (1,-1)$. Entries of $Z$ are iid uniform on {$-1,1$}. Then $X = 2$ implies $Y = 0$, such that the variables are not independent.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.