So I am trying to generalize the Isserlis' Theorem to functions that have their pdf $p(t) \propto \exp(-t^T\Gamma t)$. I am trying to imitate the proof shown here on page 28 of the pdf and 22 of the book.
I found the moment generating function to be $\exp(t^T \Gamma t)$, but I am not sure if it is correct. Please help me verify.
If it is correct, can someone also give me some hints on how to construct the corresponding lemma? Thanks in advance.
$\newcommand{\e}{\operatorname{E}}$You have $\e X=0\in\mathbb R^{n\times1}$ and $\operatorname{var}X= \Gamma\in\mathbb R^{n\times n}.$
Therefore, for $t\in\mathbb R^{n\times 1},$ you have $\e(t'X) = t'0 = 0\in\mathbb R^1$ and $\operatorname{var}(t'X) = t'\Gamma t\in\mathbb R^1.$
So $$ M_X(t) = \e(e^{t'X} ) = \e(e^{1\cdot t'X}) = M_{t'X}(1). $$ Thus the moment-generating function of the random vector $X$ evaluated at a vector $t\in\mathbb R^{n\times1}$ is equal to the moment-generating function of the scalar-valued random variable $t'X$ evaluated at $1,$ and that scalar-valued random variable has expected value $0$ and variance $t'\Gamma t.$