I need to numerically evaluate/approximate high-order moments of high-dimensional Gaussian measures/distributions with given mathematical expectations and covariance matrices. The Gaussian measures have dimension $ d$ with say $ d>1000$ and the moments have order $ k$ with say $ k>>1000$ .
Hence the classical Wick-Isserlis theorem/formula looks useless because it would involve an astronomically large number of terms.
So, please, what are my best options in order to numerically evaluate those moments in general?
My covariance matrices have a special stucture, they are pentadiagonal. Does it help in evaluating the moments?
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