Geometric embedding of random variables

Question

Given centered random variables $X_i \in \mathbb{R}$, $i=1,2,\ldots,n$ find $x^{(i)} \in \mathbb{R}^n$ such that $\langle x^{(i)}, x^{(j)} \rangle =E(X_i X_j) $ for all $i,j$.

Answer 1

We will use the following fact if $ f: \mathbb{R}^m -> \mathbb{R}^{n \times n} $ is integrable (entrywise) and $f(t)$ is positive semidefinite for all $t \in \mathbb{R}^m $ . Then $ \int_{\mathbb{R}^m } f $ is positive semidefinite.

Now let g be the multivariate distribution function on $ \mathbb{R}^n $ for $ X_1,X_2,..,X_n $ Then for any $ t \in \mathbb{R}^m $ define $[f(t)] _{i,j} = t_i t_j$ Then as $f(t)= t^t t$ it is positive semidefinite. Note $[ \int_{\mathbb{R}^m } f ]_{i,j} = E(X_i X_j) $ Then as $ \int_{\mathbb{R}^m } f $ is PSD it can be written as $ \begin{pmatrix} x^{(1)} \\ x^{(2)} \\ ... \\ x^{(n)} \end{pmatrix} $ as desired.