Suppose $X \in \{0,1\}^{n \times m}$ is a matrix generated according to the following generative process:
$$Z\_{ij} \sim \text{Bernoulli}(p) \implies X\_{ij} = \frac{Z\_{ij}}{\sum\_{k=1}^m Z\_{ik}}.$$


Is there a name for the distribution of $X$? Is there a closed-form for $E[XX^\top]$? I am struggling to find any information on the behavior of this random matrix.



Moments of rescaled Bernoulli random matrix
It is apparently assumed that the $Z\_{ij}$'s are independent, as we will do here -- since otherwise hardly anything can be said. Suppose also that $m\ge2$ and $0<p<1$.


The $ab$-entry of the matrix $Y:=XX^\top$ is
\begin{equation}
 Y\_{ab}=\sum\_{r\in[m]}X\_{ar}X\_{br}
 =\sum\_{r\in[m]}\frac{Z\_{ar}Z\_{br}}{\sum\_{k\in[m]}Z\_{ak}\sum\_{k\in[m]}Z\_{bl}},
\end{equation}
where $[m]:=\{1,\dots,m\}$ and $a,b$ are in $[n]$; here we assume the convention $\frac00=0$.
So,
\begin{equation}
 EY\_{ab}=m \,E\frac{Z\_{a1}Z\_{b1}}{\sum\_{k\in[m]}Z\_{ak}\sum\_{k\in[m]}Z\_{bl}}.
\end{equation}
If $a\ne b$ then
\begin{equation}
\begin{aligned}
EY\_{ab}&=m\,E\frac{Z\_{a1}}{\sum\_{k\in[m]}Z\_{ak}} \, E\frac{Z\_{b1}}{\sum\_{k\in[m]}Z\_{bl}} \\ 
&=m \Big(E\frac{Z\_{a1}}{\sum\_{k\in[m]}Z\_{ak}}\Big)^2 \\ 
&=m p^2\Big(E\frac1{1+\sum\_{k\in[m]\setminus\{1\}}Z\_{ak}}\Big)^2 \\ 
&=m p^2\Big(E\frac1{1+B\_{m-1,p}}\Big)^2 \\ 
&=\frac{\left(1-q^m\right)^2}{m}, 
\end{aligned}
\end{equation}
where $q:=1-p$ and $B\_{m-1,p}$ is a binomial random variable with parameters $m-1,p$.
Similarly, the diagonal entries of the matrix $EY$ are
\begin{equation}
\begin{aligned}
EY\_{aa}&=m E\Big(\frac{Z\_{a1}}{\sum\_{k\in[m]}Z\_{ak}}\Big)^2 \\ 
&=m p\,E\Big(\frac1{1+\sum\_{k\in[m]\setminus\{1\}}Z\_{ak}}\Big)^2 \\ 
&=m p\,E\Big(\frac1{1+B\_{m-1,p}}\Big)^2 \\ 
&=mp q^{m-1} \, \_3F\_2\left(1,1,1-m;2,2;-\frac{p}{q}\right), 
\end{aligned}
\end{equation}
where $\_3F\_2$ is the hypergeometric function.


