# Sample instances of random process given all temporal correlation functions?

I asked this question on signal processing stack exchange (question) but I wonder the general answer I am seeking makes the question better suited here.

Suppose I have a complex valued random process $$X[n]$$ with $$N$$ samples.* Say that all of the $$m$$-time correlation functions

\begin{align} &\langle X[i]\rangle\\ &\langle X[i]X[j]\rangle\\ &\langle X[i]X[j]X[k]\rangle\\ &\vdots \end{align}

are known for all $$i,j,k \in \{1,\ldots N\}$$. Here $$\langle X[i]X[j]\rangle$$ is the 2-time correlation function while $$\langle X[i]X[j]X[k]\rangle$$ is the 3-time correlation function. Suppose everything up to the $$N$$-time correlation function is known.

My questions are:

1) Is it possible to "simulate" or sample instances of this random process such that after enough runs of the simulation one would re-observe all of the known correlation functions? If so how could this be done? What conditions are necessary?

2) What if we don't have all $$m$$-time correlation function up to $$m=N$$ but we only have up to $$m=k? Can we still come up with a simulated random process that gives the correct statistics? It seems in this case the answer should be yes but the distribution may not be unique. For example, there are many different distributions which have the same mean and variance but different statistics nonetheless.

My thoughts: It seems to me that this is related to the moment problem. Abstractly each instance of the random process can be thought of as a point in $$\mathbb{C}^N$$ and each point in $$\mathbb{C}^N$$ is given some probability of occuring. Thus, given all moments in this higher dimensional event space I am asking the question of how to back out the probability density function. Is this the correct way to approach this?

I am happy to assume that I KNOW the moments do in fact come from some probability distribution so existence should be covered. I'm also happy with numerical approaches.

*If possible I would be interested in solutions that work continuous $$X(t)$$ with $$t\in [0,T]$$ as well.

• Does this help: en.wikipedia.org/wiki/Edgeworth_series ? – user617446 Feb 14 at 11:36
• @user617446 yes that does help. Above I limited the correlation functions known to $N$-time correlation functions, but in principle there arbitrarily high order correlation functions. I think the point is (with your link) that given the information I assume above I think it should be possible to calculate the characteristic function and cumulants for the random process of interest. Then, apparently, the Edgeworth series can be used to approximate the distribution. This seems to be what I'm looking for but I will need to spend some time with it to see if it is exactly what I'm looking for. – Jagerber48 Feb 15 at 5:23