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June 2000 Completely Operator Semi-Selfdecomposable Distributions
Makoto MAEJIMA, Ken-iti SATO, Toshiro WATANABE
Tokyo J. Math. 23(1): 235-253 (June 2000). DOI: 10.3836/tjm/1255958818

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.

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Makoto MAEJIMA. Ken-iti SATO. Toshiro WATANABE. "Completely Operator Semi-Selfdecomposable Distributions." Tokyo J. Math. 23 (1) 235 - 253, June 2000. https://doi.org/10.3836/tjm/1255958818

Information

Published: June 2000
First available in Project Euclid: 19 October 2009

zbMATH: 0985.60014
MathSciNet: MR1763515
Digital Object Identifier: 10.3836/tjm/1255958818

Rights: Copyright © 2000 Publication Committee for the Tokyo Journal of Mathematics

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Vol.23 • No. 1 • June 2000
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