Random continued fractions and inverse Gaussian distribution on a symmetric cone
First the author introduces the Wishart distribution and the inverse Gaussian distribution on the set \(H^ +_ n(R)\) of all \((n,n)\) real symmetric positive definite matrices. Then she establishes the next convergence criterion for continued fractions formed with \(H^ +_ n(R)\)-valued random variables. Theorem. Let \((Y_ k)_{k \geq 1}\) be a sequence of independent \(H^ +_ n(R)\)-valued random variables such that \({\mathcal L}(Y_{2m-1}, Y_{2m}) = {\mathcal L}(Y_ 1) \otimes {\mathcal L}(Y_ 2)\), \(m \geq 1\). Assume that \({\mathcal L}(Y_ 1)\) and \({\mathcal L}(Y_ 2)\) are absolutely continuous with respect to the Lebesgue measure, and that their support is \(H^ +_ n(R)\). Then the sequence \(([Y_ 1, \dots, Y_ k])_{k \geq 1}\) converges almost surely, where \([Y_ 1] = Y_ 1\) and \([Y_ 1, \dots, Y_ m] = Y_ 1 + [Y_ 2,\dots, Y_ m]^{-1}\), \(m \geq 2\). All results concerning \(H^ +_ n(R)\) are paralleled by results in which the cone \(H^ +_ n(R)\) is replaced by an abstract irreducible symmetric cone. Unfortunately, the paper is somewhat poorly written.
convergence criterion for continued fractions
scientific article; zbMATH DE number 757596
scientific article