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A comment on Gaussian Processes, not quite for dummies. Parent comment
This is not quite strong enough. You need (X_t1, X_t2, ..., X_tn) to jointly have a multivariate Gaussian distribution for every finite (t1, ..., tn).
Also, for correlation, need to assume the expectations exist.
This is not quite strong enough. You need (X_t1, X_t2, ..., X_tn) to jointly have a multivariate Gaussian distribution for every finite (t1, ..., tn).