I am not a professional mathematician so please excuse me if my question is not phrased correctly.

I am interested in the following simple sounding problem.

Consider a random $n$ by $n$ $0$-$1$ matrix $M$ where $M_{i,j} = 1$ with probability $1/2$ and $0$ otherwise. Now choose a single random $0$-$1$ vector $v$ where $v_i=1$ with probability $1/2$. I can write down a precise formula for the probability distribution of $u=Mv$. However, I would like to argue that for large $n$ the distribution of $u_i$ becomes very close (perhaps converges to in some sense) to i.i.d. $Bin(n,1/4)$.

My hand wavy intuition which might also reveal my motivation a bit more is that each $u_i$ is tightly concentrated around its mean (as is the number of $1$s in $v$ and each row of $M$). Under these circumstances knowledge of $u_i$ tells you almost nothing about $v$ and hence almost nothing about other $u_j$. In other words the $u_i$ are in some sense very close to being independent with high probability.

Is there some way in which this can be formalized?

Here is an example of what I would like to do. Say we have a set $S$ of $2^n$ $0$-$1$ vectors chosen i.u.d. and for each we compute the product $Y_i=MS_i$ (I have overloaded the notation as here $S_i$ and $Y_i$ refer to the $i$th vector not the $i$th element within a vector). Let us make the matrix $M$ $m$ by $n$, that is potentially non-square, but still $0$-$1$ and each $M_{i,j}$ is i.u.d. as before. I would like to work out the expected number of distinct $Y_i$ for large $n$ and $m$. The number of rows $m$ will typically be smaller than $n$ but not by too much. I would like to be able to argue as if each element of $Y_i$ were independent to make the math easier. I only care about large $n,m$ approximations.

jointlaw of the coefficients it is difficult to say much... – UwF Mar 30 at 8:44