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Given a random process $x[n] \in \mathbb{R}$ (say of length $N$) and all correlation functions such as:

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

  1. Is it possible to simulate a single realization of the random process and if so how? Thinking abstractly about the problem it seems like any trajectory could be mapped onto a vector in $\mathbb{R}^n$ and there is some probability distribution on $\mathbb{R}^n$. Sampling this distribution and then picking out the corresponding trajectory would then constitute of "simulation" of a single realization that I am looking for. It seems like given information about all of the auto-correlation functions it should be possible to determine the probability distribution. This seems related to the inverse moment problem.
  2. What if we truncate and only have up to $k$-time correlation function with $k<N$? My guess is that it is not possible to uniquely generate the probability distribution but it should be possible to (non-uniquely) simulate a process with matching auto-correlation functions up to $k$.
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Here is one possible method that you could possibly use.

Let's model your random process $x[n]$ as an auto-regressive (AR) process of order $P$, that is, $$x[n] + \sum_{p=1}^P a_px[n-p] = c + \epsilon[n]$$ where $c$ is some constant that I will set to 0 in what follows for the sake of simplicity, $\{a_p\}_{p=1}^P$ are the parameters of the AR model and $\epsilon[n]$ is some white noise of variance $\sigma^2_\epsilon$.

Multiplying each side of the above equation by $x[n-l]$ for some $l \ge 0$ leads us to $$\left(x[n] + \sum_{p=1}^P a_px[n-p]\right)x[n-l] = \epsilon[n]x[n-l].$$ Now, taking the expectation on each side, we obtain the Yule-Walker equations $$C_x[l] + \sum_{p=1}^P a_pC_x[l-p] = \begin{cases} 0 & \text{if} & l > 0, \\ \sigma^2_\epsilon & \text{if} & l = 0 \end{cases}$$ using the fact that $\epsilon[n]$ and $x[n-l]$ are uncorrelated for $l > 0$ (you can see that from the equation of the AR process above). Using this equation with $l = 0, 1, 2, \dots, P$, you have $P+1$ equations to solve for the $P + 1$ unknowns (i.e., $\{a_p\}_{p=1}^P$ and $\sigma^2_\epsilon$).

You then have a model to generate a time series given the temporal auto-correlation.

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  • $\begingroup$ thank you for your answer. It looks like this answers my question in the case that the stochastic process can be modeled as an AR process. I guess this does not work for more complicated processes, for example a process which follows a non-linear stochastic differential (difference) equation. Is this correct? I wonder if there is an approach that would work for a fully general random process. $\endgroup$ – Jagerber48 Feb 5 at 17:50
  • $\begingroup$ I'm not aware of the existence of a "general approach" but I'm not an expert in that field. $\endgroup$ – anpar Feb 6 at 8:01
  • $\begingroup$ Still, I believe my answer answers your question. You might want to ask a new question for the more general case you describe in the above comment. $\endgroup$ – anpar Feb 11 at 20:15
  • $\begingroup$ I've asked the question on mathematics stack exchange asking for the more general approach: math.stackexchange.com/questions/3104827/… . There has been no attention there yet. $\endgroup$ – Jagerber48 Feb 11 at 22:03

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