## Completely operator semi-selfdecomposable distributions.(English)Zbl 0985.60014

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.

### MSC:

 6e+08 Infinitely divisible distributions; stable distributions 6e+06 Probability distributions: general theory

Zbl 0947.60010
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