Multivariate reciprocal inverse Gaussian distributions from the Sabot -Tarr\`es -Zeng integral

In Sabot and Tarr`es (2015), the authors have explicitly computed the integral $$ST{Z}_n=\int \exp (-⟨x,y⟩)(\det M_x{)}^{-1\mathrm{/}2}dx$$ where $M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonal coefficients and with diagonal $(2x_1,\dots ,2x_n)$. The domain of integration is the part of ${\mathbb{R}}^n$ for which $M_x$ is positive definite. We calculate more generally for $b_1\ge 0,\dots b_n\ge 0$ the integral $$GST{Z}_n=\int \exp (-⟨x,y⟩-\frac{1}{2}{b}^{\ast }{M}_x^{-1}b)(\det M_x{)}^{-1\mathrm{/}2}dx,$$ we show that it leads to a natural family of distributions in ${\mathbb{R}}^n$, called the $GST{Z}_n$ probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. We also show that if the graph with the set of vertices $V=\{1,\dots ,n\}$ and the set $E$ of edges $\{i,j{\}}^{\mathrm{\prime }}$ s of non zero entries of $M_x$ is a tree, then the integral $$\int \exp (-⟨x,y⟩)(\det M_x{)}^{q-1}dx$$ where $q>0,$ is computable in terms of the MacDonald function $K_q\mathrm{.}$