Assume $A$ is a gaussian matrix of size $n \times N$ where $n\ll N$. All the entries of $A$ are independent and identically distributed normal $\mathcal N(0, 1/n)$.
Moreover, assume that $x$ is a vector of size $N$.
Denote $H=A^{*}A-I$, where $I$ is the identity matrix. Can someone help me understand the distribution of $Hx$?
In a research paper I read (pg 3, section D), they state that $Hx$ behaves as a kind of noisy random vector and is accurately modeled as a vector with i.i.d. Gaussian entries with variance $ \frac{1}{n} \|x\|^{2}$
Can someone help me prove it?
I also have a follow-up question:
What can I say about distribution of $Hx$, if all entries of $A$ were independent and identically distributed $\operatorname{Bern}(p)$ variables. .