For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B\_1, \ldots, B\_n)$ with correlation $C$ ?


Following the approach described in this MO [thread](https://mathoverflow.net/questions/18268/discrete-stochastic-process-exponentially-correlated-bernoulli), one can think of the following construction. Define independent Bernoulli random variables $B\_{k\_1, \ldots, k\_n}$ for $(k\_1, \ldots, k\_n) \in \mathbb{Z}^n$ and another independent $\mathbb{Z}^k$-valued random variable $I=(I\_1, \ldots, I\_n)$. Then $(B\_{I\_1}, \ldots, B\_{I\_n})$ is a correlated Bernoulli vector.


1: Is there any example of correlation structure that cannot be obtained this way ?


2: Any easy example of correlation matrix $C$ that cannot be the correlation matrix of a $\{0,1\}^n$ valued random vector ?



Constructing Bernoulli random variables with prescribed correlation
Here's a pretty general construction. Take unit vectors $v\_1,\dots,v\_n$ in $\mathbb{R}^n$ and let $u$ be a random unit vector (chosen with the uniform probability measure on the unit sphere). Define $B\_i$ to be 1 if the inner product of $u$ and $v\_i$ is positive and -1 otherwise. Then the correlation between $B\_i$ and $B\_j$ is the inner product of $v\_i$ and $v\_j$. (I haven't checked that carefully but I think it's true.)


Added in the light of the comment below: OK, I should have checked. It's actually not the inner product but π minus twice the angle between the two vectors all over π. I.e., it depends linearly on the angle between the two vectors, is 1 when that angle is zero and -1 when it is π. The angle is the inverse cos of the inner product, which gives us a formula. 


So it gives us a fairly big supply of matrices -- I can't quite decide whether it gives us all (or rather all for which the variables take two values, each with probability 1/2).


A more general-looking construction is this. Take any probability space and let $A\_1,...,A\_n$ be sets of measure 1/2. Pick a random point x and let $B\_i$ be 1 if x is in $A\_i$ and -1 otherwise. But that becomes trivial, because if you have any set of Bernoulli variables taking the values $\pm 1$ with probability 1/2, then you can set $A\_i$ to be the set where $B\_i=1$.


