Products of Gaussian stochastic process variables

Question

In the classic experimental physics text "Statistical Theory of Signal Detection" by Carl. W. Helstrom, Chapter II, section 4 concerns Gaussian Stochastic Processes. Such a process is observed at times $t_1, t_2, t_3, ... t_n$ to obtain $n$ random variables $x_1, x_2, x_3, ... x_n$, then the probability density for the $n$ variables is of the form:
$$p_n(x_1,t_1; x_2,t_2; ...; x_n,t_n) = M_n\;\exp(-0.5\Sigma_j\Sigma_k\mu_{jk}x_jx_k).$$
where $\mu_{jk}$ is a positive definite matrix and $M_n$ is a normalization constant to give unit probability when you integrate over all possible values:
$$M_n = (2\pi)^{-n/2}\;|\;\textrm{det}\;\mu|^{0.5},$$ where $\mu$ is the determinant of the matrix $\mu_{jk}$. We assume that the expected values are all zero: $E(x_k)=0$.

He notes that the expected value of the product of any odd number of $x_j$ is zero (which seems to follow from symmetry), and gives a formula for the expected value of a product of even numbers of variables. We define $\phi_{jk}$ as:
$$\phi_{jk} = E(x_jx_k).$$ He then notes that:
$$E(x_1x_2x_3x_4) = \phi_{12}\phi_{34} + \phi_{13}\phi_{24} + \phi_{14}\phi_{23}.$$

Is there a simple proof? And is there a simple proof that relates to the methods of quantum mechanics / quantum field theory?

Answer 1

Yes, it is simple to prove using moment generating functions. And yes, the mathematics is very closely related to that of quantum field theory.

You compute $G(j) = <exp(\sum j_i x_i)>$ where each $j_i$ is a "source" for the corresponding $x_i$. This is easily shown to be something like $G(j) = exp(\sum j_i \mu_{ij}^{-1} j_j)$ To get expectation values you then take $ <x_i x_j ...> = \frac{\partial}{\partial j_i} \frac{\partial}{\partial j_j} ... G(j)|_{j=0}$ . The rest follows simply. In particular, you can see how the variables must be grouped in pairs to get a nonzero results when you set $j=0$ after taking derivatives. You essentially have a Feynman expansion of a non-interacting 0-dimensional field theory.

This is covered nicely in Zee's "Quantum Field Theory in a Nutshell", where it is a simple application of what he calls the "central identity of quantum field theory"

Answer 2

This might be slightly late, but I just wanted to slightly touch on a few more things.

For the Gaussian case, there is a very nice theorem on reducing high-order derivatives to a combinatorics problem (this can be done via relating the moments of the random variables and their corresponding cumulants): Wick's theorem. There's a also a generalization of this theorem known as the Isserlis' theorem. These techniques arise in perturbative QFT.