Let $W$ be a $n\times m$ matrix with i.i.d. $\mathcal{N}(0, 1)$ entries. If $x\in \mathbb{R}^m$ is a vector, then $Wx\in \mathbb{R}^n$ is a random vector. Since the linear combination of independent normal ranodm variables is normal, it follows that $Wx$ is a vector of independent $\mathcal{N}(0, ||x||_2^2)$ entries.
Now I am in a situation where $x$ is itself a random vector but chosen independently of $W.$ I am interested in understanding the distribution of $Wx.$ Assume that $x\sim \rho$ where $\rho$ is some probability measure absolutely continuous w.r.t. Lebesgue measure. Conditioned on $x,$ $Wx$ is a vector of i.i.d. Gaussian entries with mean $0$ and variance $||x||_2^2.$ It follows that the coordinates of $Wx$ are identically distributed. Since $W$ and $x$ are independent, I think it should be possible to say that the entries of $W.x$ are independent as well. But I am not able to formally write it. If the entries are not independent, is it at least possible to write the law of $Wx$ explicitly?