Multivariate reciprocal inverse Gaussian distributions from the Sabot–Tarrès–Zeng integral

In Sabot and Tarrès (2015), the authors have explicitly computed the integral STZn=∫exp(−〈x,y〉)(detMx)−1∕2dxwhere Mx is a symmetric matrix of order n with fixed non-positive off-diagonal coefficients and with diagonal (2x1,…,2xn). The domain of integration is the part of Rn for which Mx is positive definite. We calculate more generally for b1≥0,…bn≥0 the integral ∫exp−〈x,y〉−12b⊤Mx−1b(detMx)−1∕2dx,we show that it leads to a natural family of distributions in Rn, called the MRIGn 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. In general, if the power of detMx under the integral in STZn is distinct from −1∕2 it is not known how to compute the integral. However, introducing the graph G having V={1,…,n} for set of vertices and the set E of {i,j}′ s of non-zero entries of Mx as set of edges, we show also that in the particular case where G is a tree, the integral ∫exp(−〈x,y〉)(detMx)q−1dxwhere q>0, is computable in terms of the MacDonald function Kq.