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

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Abstract

In Sabot and Tarrès (2015), the authors have explicitly computed the integral STZn=exp(x,y)(detMx)12dxwhere 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 b10,bn0 the integral expx,y12bMx1b(detMx)12dx,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 12 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)q1dxwhere q>0, is computable in terms of the MacDonald function Kq.

AMS subject classifications

primary
60E05
secondary
62 E10

Keywords

Laplacian of a graph
MacDonald function
Multivariate reciprocal inverse Gaussian
Supersymmetry

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