Given a stochastic process $X:S\times \Omega\rightarrow \Bbb{R}$ with $S\subset \Bbb{R}^d$, one can define its moment functions as follows (when they exist):
$$ M_{\boldsymbol{\alpha}}(t_1,\ldots,t_n) = \Bbb{E}[X(t_1)^{\alpha_1}\cdots X(t_n)^{\alpha_n}] ,\quad t_1,\ldots,t_n\in S $$ If one knows the first and second order moments, $\mu(t)$ and $c(t_1,t_2)$, there exists a process with these moments, in particular a Gaussian process. What if I know higher order moments? Is there a standard construction that produces a process $X$ with a given moment function sequence?