Example of Darmois Skitovitch theorem In looking at something, I stumbled across the awesome Darmois-Skitovitch theorem:
Define two random variables $y_1$ and $y_2$ as linear combinations of independent random variables $s_i$ $(i=1, · · · , q)$:
$$y_1=\sum_{i=1}^q\alpha_i s_i$$
$$y_2=\sum_{i=1}^q\beta_i s_i$$
Then, if $y_1$ and $y_2$ are independent, all variables $s_j$ for which $a_jb_j \neq 0$ are Gaussian.
Although not explicit in the theorem, this seems to indicate that if some of the $s_j$ are Gaussian, you can have some of those $a_jb_j \neq 0$ and have $y_1$ and $y_2$ be independent. I tried to create some simple examples of this, but have not been successful. It is easy to do if the $s_j$ are not independent, but that's not what I'm looking for. Anyone have a good example (or way of looking at it)?

Let $X_1, X_2$ be iid $\mathcal{N}(0,1)$
Then $Y_1 = X_1 + X_2$ and $Y_2 = X_1 - X_2$ are independent.
This is also an example of the stricter Kac-Bernstein theorem.
Example: Throw two darts at a 1D target. Each throw is independent. If the distance between the two darts and the summed distance of the two darts from the center are independent then the distance of each dart from the center is a Gaussian variable.
