Independence of two random variable

Question

Let $W$ and $S$ are two positive valued continuous random variable. Suppose $g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ is a constant.

Is it possible, there exist some $\theta>0$ such that the following two statements holds simultaneously,

[i] $W$ is independent of $SW+S\theta$, and

[ii] $W$ is independent of $SW+Sg(S)$ ??

Answer 1

Take any real $\theta>0$ and any independent positive continuous random variables $W$ and $X$. Let $S:=X/(W+\theta)$ and $g(x):=\theta$ for all $x$. Then $SW+S\theta=SW+Sg(S)=X$, so that $W$ is independent of $SW+S\theta$ and of $SW+Sg(S)$.