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Suppose $X$ is a multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by its multiplication with some other random variable $Y$, i.e., $\eta = YX$.

What should I need to require from $Y$ in order that $\eta$ will be Gaussian but will also be uncorrelated with $X$?

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Take a Cauchy distribution for $Y$: define $Y=Z/X$ with $Z$ a zero-mean Gaussian independent of $X$; then $\eta=YX=Z$ has the desired property of having a Gaussian distribution and being uncorrelated with $X$.

The answer becomes more complicated if $X$ has nonzero mean, as explained here.

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Without much loss of generality let‘s assume that $X$ and $\eta$ are univariate standard normally distributed variables. In order to achieve independence we need to require the following $$ \mathrm{Cov}(\eta, X)=\mathbb{E}[YX^2]=\mathbb{E}[Y]=0. $$ And for $\eta$ to be Gaussian it needs to hold that $$ \phi_\eta(t)=\mathbb{E}_Y[\phi_X(tY)]=\mathbb{E}_Y\Big[\exp\Big(-\frac{(tY)^2}{2}\Big)\Big]=\exp(-t^2/2), $$ (unfortunately, I wasn’t able to simplify further). If you need only one example of distribution of $Y$, then a fair coin with values +1 and -1 will work. In order to extend to multivariate distribution, consider a vector of such $Y$‘s.

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