# Existence of independent linear combinations of random variables

Let $$X,Y$$ be independent multivariant random variables on $$\mathbb{R}^d$$. Let $$\alpha,\beta$$ be two positive real values such that $$\alpha^2+\beta^2=1$$. Then, $$S=\alpha X+\beta Y$$ is a new random variable. The question is: do there exist two real values $$a,b$$, such that $$T=aX+bY$$ is independent with $$S$$? If so, are there closed-form expressions of $$a$$ and $$b$$?

• In case $X,Y$ are jointly normal, then $S,T$ are independent if and only if they are orthogonal. Apr 16, 2021 at 20:22
• If $S,T$ are independent then each component of $S$ and $T$ are independent, hence uncorrelated. For the case $d = 1$ you get for the covariance $C(S,T) = \alpha a V(X) + \beta b V(Y)$ and this can be easily solved for $a$ and $b$. If the variances of the components of $X$ and $Y$ are not identical, these solutions for $a$ and $b$ will differ. Thus for the case $d > 1$ there seem to exist no solutions except in trivial cases. Apr 16, 2021 at 22:19