There are N jointly distributed Gaussian random variables, complex in nature, and are uncorrelated. How to prove, they are statistically independent?

Question

I came across this as a property, but wanted to prove it myself, but can't get through with the proof. Any help would be appreciated.

Answer 1

Is this true?

Suppose we let $Z = X + i Y$, with $X,Y \sim N(0,1)$ independent and Normally distributed, and let $W_n$ be i.i.d. draws from a uniform distribution on $\{-1,1\}$, i.e.

$$\mathbf P[ W_n = 1] = \mathbf P[W_n = -1] = \frac12,$$

(often referred to as the Rademacher distribution). Now define

$$Z_n = W_n Z, \qquad n = 1,\ldots, N.$$

Clearly the $Z_n$ are not independent, since for example

$$ \mathbf P[ Z_n = \pm z \, | \, Z_1 = z ] = 1, \qquad n = 1,\ldots, N.$$

And then computing the covariance we have for any $m \neq n$

\begin{align*} \text{Cov}(Z_m,Z_n) & = \mathbf E[ Z_m Z_n ] - \mathbf E[Z_m] \mathbf E[Z_n] \\ & = \mathbf E[ Z_m Z_n] \\ & = \mathbf E[ W_m W_n Z^2] \\ & = \mathbf E[W_m] \mathbf E[W_n] \mathbf E[Z^2] \\ & = 0, \end{align*} where in the first line we used the fact that $\mathbf E[Z_n] = 0$, and in the last line we used independence of the $W_n$, and the fact that $\mathbf E [W_n] = 0$.