Dependence between summation and difference of normal random variables

Question

Suppose $x,y$ i.i.d. with distribution $\mathcal{N}(0,\sigma ^2) $. Then $z=x+y$ and $w=x-y$. The marginal distribution for $z\text{ and }w$ is $\mathcal{N}(0,2\sigma ^2)$. How can we prove that $w$ and $z$ are independent or not independent?

Answer 1

We define the vector $U=(z,w)$.

Let $(\lambda$, $\beta) \in \mathbb{R^2}$, we have that $\lambda z+\beta w=(\lambda+\beta)x+(\lambda-\beta)y$, therefore U is a Gaussian vector because x and y are independent and normally distributed, hence $\lambda z+\beta w$ is Gaussian too.

The independence can be proved by showing that $z$ and $w$ are uncorrelated. $cov(z,w)=var(x)-var(y)$, and because $var(x)=var(y)$, we can conclude that z and w are independent.