When are S and T uncorrelated based on the marginal distributions of X and Y? With S = X - Y and T = X + Y

Question

Given two random variables $X$ and $Y$, with $S = X - Y$ and $T = X + Y$. Under what constraint on the marginal distributions of $X$ and $Y$ are $S$ and $T$ uncorrelated.

I know that $S$ and $T$ are uncorrelated if the $Cov(S, T) = 0$.

If two variables $S$ and $T$ are independent their covariance has to be 0. So now I want to know in what cases $S$ and $T$ are independent based on the marginal distributions of $X$ and $Y$.

Answer 1

No, you do not need to know whether $S$ and $T$ are independent. Just concentrate on the covariance. Expand out $\text{Cov}(X-Y, X+Y)$ using the properties of covariance and you'll see the answer.

Answer 2

Based on the properties of the covariance, one gets the desired relation between $X$ and $Y$:

\begin{align*} \text{Cov}(S,T) & = \text{Cov}(X - Y,X + Y)\\\\ & = \text{Cov}(X,X) + \text{Cov}(X,Y) - \text{Cov}(Y,X) - \text{Cov}(Y,Y)\\\\ & = \mathbb{V}(X) - \mathbb{V}(Y) = 0 \end{align*}