I am working on Gaussian comparison theorem. My question is as follows.
Question. Suppose that $X$ and $Y$ are two independent centered gaussian random variables. Let $Z=\sqrt{1-t}X+\sqrt{t}Y$ for all $t\in [0,1]$. And let $\mathbb{E}[X^2]\leq \mathbb{E}[Y^2]$. Then is it ture that there exists $a> 0$ and a random vector $W$ that is independent of $Z'$ such that $Z=aZ'+W$? where $Z'$ is a $t$ derivative of $Z$.
The related theorem is as follows. Please see the last paragraph of the page.
Take $W=X$ and $\alpha=0$. This satisfies your requirements.
Answer for the revised version of the question:This is not always possible. Take $Y=X$. There is no $\alpha >0$ such that $X-\alpha X$ is indepndent of $\alpha X$.