Completely operator semi-selfdecomposable distributions
In a previous paper [ibid. 22, No. 2, 473-509 (1999; Zbl 0947.60010)] the authors have introduced operator semi-selfdecomposable distributions. Let \(0<b<1\) and \(Q\) a \(d\times d\) matrix whose eigenvalues have positive real part. \(L_{-1}(b,Q)\) is defined as the class of all infinitely divisible probability measures on \({\mathbb R}^d\). For \(m=0,1,\ldots\), a probability measure \(\mu\) on \({\mathbb R}^d\) belongs to \(L_m(b,Q)\), if there exists a probability measure \(\rho\in L_{m-1}(b,Q)\) and a pair of independent radom variables \(X\) and \(Y\) with distributions \({\mathcal L}(X)=\mu\) and \({\mathcal L}(Y)=\rho\) such that \(X \buildrel \text{(d)} \over = b^QX + Y\), where \(\buildrel \text{(d)} \over = \) denotes equality in distribution. In this paper the distributions in \(L_\infty(b,Q)=\bigcap_{m\geq 0} L_m(b,Q)\) (called completely operator semi-selfdecomposable) are characterized in terms of their Gaussian covariance matrices and their Lévy measures. Furthermore, it is shown that \(L_\infty(b,Q)\) is the smallest class closed under (weak) convergence, \(Q\)-type equivalence, convolution, and under going to the \(t\)th convolution for any \(t>0\), that contains the class \(\text{OSS}(b,Q)\) of operator semi-stable distributions.
scientific article; zbMATH DE number 1523380