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Assume the variable $X_t$ has a continuous value, sampled over uniform discrete time (time series).

  1. I have the full autocorrelation function up to max time lag $l_{max}$. The coefficients may be both positive and negative. Some coefficients may be zero.
  2. I have the full time series "history", $X_0$ to $X_t$ necessary to satisfy any initial lag conditions during the beginning of the sequence generation. I'd like to generate any number of random possible sequence extensions $X_{t+1}$ to $X_{t+N}$, where the extension sequence fully honors (or at least converges on, with some reasonable tolerance) all autocorrelation coefficients of the historical period up to max time lag $l_{max}$.
  3. The distribution of the data is complex and multimodal, so I would prefer a non-parametric method, if possible, however I realize this could further complicate things. So if a non-parametric technique is too difficult, I am willing to compromise with sampling from a loosely fitting gaussian distribution. Or if it would simplify things, we could also assume I have a KDE estimate of the distribution available to sample from: ${\widehat {f}}_{h}(X)$

Any ideas or algorithms are welcome, including iterative techniques. Thanks in advance.

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  • $\begingroup$ Does it matter that $X_t$ is continuous if it is sampled over uniform discrete time? If not, I think you could try simulating from a moving average (MA) model, since this model can be defined by its autocorrelations. If you have $l_{max}$ of them, it would be an MA($l_{max}$) model. $\endgroup$ Commented Aug 26, 2024 at 9:21

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Did you have a look at block bootstrap?

Block bootstrap assumes you have a sample of the random process $X_t$ (which obviously has the autocorrelation you seek, then you can just resample (draw with replacement) blocks of a length longer than the maximal autocorrelation you are interested. The resulting resamples should have an estimated autocorrelation very close to the one you have in your original sample.

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  • $\begingroup$ This is an interesting proposal, but is there any scheme to "transition" the blocks in a seamless fashion? Remember $X_t$ is a continuous value, so just piecewise concatenation of random blocks in themselves would tend to cause obvious and abrupt jumps. $\endgroup$ Commented Aug 28, 2024 at 0:25
  • $\begingroup$ I think people introduced smooth block bootstrap versions $\endgroup$ Commented Aug 28, 2024 at 5:27
  • $\begingroup$ Could you please edit your answer to provide a reference to the "smooth block bootstrap"? If there are also other variations, it might be nice to also mention the more popular ones. $\endgroup$ Commented Aug 28, 2024 at 13:35

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