Question

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

Answer 1

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.