## Tokyo Journal of Mathematics

- Tokyo J. Math.
- Volume 23, Number 1 (2000), 235-253.

### Completely Operator Semi-Selfdecomposable Distributions

Makoto MAEJIMA, Ken-iti SATO, and Toshiro WATANABE

#### 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.

#### Article information

**Source**

Tokyo J. Math., Volume 23, Number 1 (2000), 235-253.

**Dates**

First available in Project Euclid: 19 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.tjm/1255958818

**Digital Object Identifier**

doi:10.3836/tjm/1255958818

**Mathematical Reviews number (MathSciNet)**

MR1763515

**Zentralblatt MATH identifier**

0985.60014

#### Citation

MAEJIMA, Makoto; SATO, Ken-iti; WATANABE, Toshiro. Completely Operator Semi-Selfdecomposable Distributions. Tokyo J. Math. 23 (2000), no. 1, 235--253. doi:10.3836/tjm/1255958818. https://projecteuclid.org/euclid.tjm/1255958818