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

TL;DR

This paper introduces a new family of multivariate distributions called GSTZ_n, derived from integrals related to reciprocal inverse Gaussian distributions, with properties like stability under marginalization and conditioning.

Contribution

It generalizes the Sabot-Tarrès-Zeng integral to define the GSTZ_n family, revealing their properties and connections to known functions like the MacDonald function.

Findings

GSTZ_n distributions are stable under marginalization and conditioning.

The integral over tree graphs can be expressed using MacDonald functions.

The family extends properties of univariate reciprocal inverse Gaussian distributions.

Abstract

In Sabot and Tarr\`es (2015), the authors have explicitly computed the integral $$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$ where $M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonal coefficients and with diagonal $(2x_1,\ldots,2x_n)$. 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\geq 0,\ldots b_n\geq 0$ the integral $$GSTZ_n=\int \exp \left(-\langle x,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$ we show that it leads to a natural family of distributions in $\mathbb{R}^n$, called the $GSTZ_n$ 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 $V=\{1,\ldots,n\}$ and the set $E$ of edges $\{i,j\}'$ s of non zero entries of $M_x$ is a tree, then the integral $$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$ where $q>0,$ is computable in terms of the MacDonald function $K_q.$