This paper mentioned a type of random noise matrix $\boldsymbol{W}\in\mathbb{R}^{m\times n}$ known as the generalized white noise matrix (definition 2.15 in the paper) which satisfies the following requirements:
- All entries of $\boldsymbol{W}$ are independent.
- Each entry $\boldsymbol{W}_{ij}$ has mean zero and variance $1/n$, and higher moments satisfying, for each integer $p\geq3$,
\begin{align} \lim_{m,n\rightarrow\infty} n\mathbb{E}\Big[|W_{ij}|^p\Big]=0. \end{align} An example is $\boldsymbol{W}_{ij}\stackrel{iid}{\sim}N(0,1/n)$.
(Note that $i\in\{1,\dots,m\}$ and $j\in\{1,\dots,n\}$.)
Question: Given that I have a random Bernoulli matrix with entries $\boldsymbol{X}_{ij}\stackrel{iid}{\sim}\text{Bernoulli}(c/n)$, where $c$ can be any constant that is independent of $m$ and $n$. Can I decompose it into two parts (one part being a constant matrix and the other part being a random matrix which can be multiplied by a constant) such that the random matrix part is a generalized white noise matrix? What is the best I can do to produce the generalized white noise matrix in the decomposition? I'm also interested in other forms of decomposition to produce the generalized white noise matrix (apart from the one that I have suggested). Essentially, I'm interested in any way I can construct a generalized white noise matrix from my Bernoulli matrix.
My attempt: The best I could do is to decompose $\boldsymbol{X}$ into a constant matrix (basically a matrix with the mean of the bernoulli distribution in all the entries) and another random matrix (multiplied with a constant; the constant is to rescale the matrix so that the variance of its entries become $1/n$). The random matrix satisfies all the requirements of the generalized white noise matrix except for $\lim_{m,n\rightarrow\infty} n\mathbb{E}\Big[|W_{ij}|^p\Big]=0$ for $p\geq3$ (note that this condition is satisfied if we were to allow $c=\Theta(\log n)$ but that would break the constant requirement of $c$).