You will probably get a better answer to this on stats.stackexchange.com/questions

@mr.gondolier: Is $f(x) = x^4$ allowed? You state in the question that $f(x)$ is bounded.

@Didier: See my comment on Darsh's answer.

It depends on what you are doing. Say you want uniformly distributed random variables with a particular correlation structure. You can't just generate uncorrelated uniform random variables and apply the Cholesky decomposition because a sum of uniform random variables is no longer uniform. It will have the correct correlation (as you show) but the marginal distributions will not be uniform.

You will probably have more luck with this question on stats.stackexchange.com.

Still, I did like Daniel's plane story :)

This is a bit off topic, but related to gowers last sentence. I don't really like the idea of having to justify that proof is important. I think asking the question Why?' is natural and important in its own right. Curiosity appears to be built in to us. 'Why?' drives a significant portion of the sciences and humanities and everyday life. It would be hard to deny that our active desire to satisfy this inbuilt curiosity is at least partially responsible for human advance (whatever that is). For me, trying to find specific examples where having a proof saves lives' cheapens the whole process.

Way to ruin the story Richard!

That Strong Law of Large Numbers paper is fantastic!

I know $K_n$ is the complete graph and $C_n$ is a cycle graph and I am assuming that $E_n$ is the empty graph. What are $A_n$ and $D_n$?

Neat idea. I think you might mean $A^{m-1}$? Also, don't you need to sum all of the elements in the matrix, not just those above and on the diagonal?

Indeed, I mean vectors. I have fixed the title. I have made a bit of a mess of this question! And multiplying Peters answer by $m!$ as you have said give the desired result.