How to properly define a space for correlation matrices?

Question

One usually defines the space $$\{M\in\mathrm{Mat}_n(\mathbb R):\forall j, m_{jj}=1,M^\top=M\ge0\}$$ to be the space of correlation matrices. It is clear that every correlation matrix is inside this space. But how to show that for each $M$ in this space it associates with a random vector $X$ whose correlation matrix is exactly $M$, i.e. how to construct such $X$?

Answer 1

I do not think you can take any random vector X, but if X is i.i.d. standard normal then a common numerical technique to obtain correlated standard normals is to multiply the matrix $\mathbf u$ of samples from X by the Cholesky decomposition of M

I believe as the sample size $n\to\infty$ the resulting correlation matrix converges to M