Il 28 maggio alle ore 14
in Aula di Consiglio del Dipartimento di Matematica, Sapienza Università di
Roma
di terrà il seguente seminario di Probabilità e Statistica Matematica:
GERARD LETAC, IMT, Université Paul Sabatier, Toulouse, France
MULTIVARIATE RECIPROCAL INVERSE GAUSSIAN DISTRIBUTIONS: THE SURPRISING
INTEGRALS OF SUPERSYMMETRY
Abstract
If $W=(w_{ij})_{1\leq i,j\leq n}\)$ is a symmetric matrix with $w_{ij}\geq
0$ and zero diagonal, consider the matrix
$M_x=2\mathrm{diag}(x_1,\ldots,x_n)-W$
and the set $C_W$ of $x\in \mathbb{R}^n$ such that $M_x$ is positive
definite. Physicists and probabilists have considered in the last ten years
several integrals equivalent to $\int_{C_W}\exp(-a^*M_xa-b^*M_x^{-1}b)(\det
M_x)^{q-1}dx$. According to the properties of the undirected graph $G$
associated to $W$ with vertices $1,\ldots,n$ and edges $(i,j)$ such that
$w_{ij}\neq 0$, we compute some of these integrals and we study the
corresponding exponential families on $\mathbb{R}^n$ with many attractive
properties.