Completely Operator Semi-Selfdecomposable Distributions

Abstract

The class $L_\infty(b, Q)$ of completely operator semi-selfdecomposable distributions on $\mathbf{R}^d$ for $b$ and $Q$ is studied. Here $0<b<1$ and $Q$ is a $d\times d$ matrix whose eigenvalues have positive real parts. This is the limiting class of the decreasing sequence of classes $L_m(b, Q)$, $m=-1,0,1,\ldots$, where $L_{-1}(b, Q)$ is the class of all infinitely divisible distributions on $\mathbf{R}^d$ and $L_m(b,Q)$ is defined inductively as the class of distributions $\mu$ with characteristic function $\hat{\mu}(z)$ satisfying $\hat{\mu}(z)=\hat{\mu}(b^{Q'}z)\hat{\rho}(z)$ for some $\rho\in L_{m-1}(b,Q)$. $Q'$ is the transpose of $Q$. Distributions in $L_\infty(b,Q)$ are characterized in terms of Gaussian covariance matrices and Lévy measures. The connection with the class $OSS(b,Q)$ of operator semi-stable distributions on $\mathbf{R}^{d}$ for $b$ and $Q$ is established.