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December 1999 Operator Semi-Selfdecomposability, $(C,Q)$-Decomposability and Related Nested Classes
Makoto MAEJIMA, Ken-iti SATO, Toshiro WATANABE
Tokyo J. Math. 22(2): 473-509 (December 1999). DOI: 10.3836/tjm/1270041450

Abstract

There are two types of generalizations of selfdecomposability of probability measures on $\mathbf{R}^d, d\geq 1$ : the $c$-decomposability and the $C$-decomposability of Loève and Bunge on the one hand, and the semi-selfdecomposability of Maejima and Naito on the other. The latter implies infinite divisibility but the former does not in general. For $d\geq 2$ introduction of operator (matrix) normalizations yields four kinds of classes of distributions on $\mathbf{R}^d : L_{0}(b,Q),\tilde{L}_{0}(b,Q),L_{0}(C,Q)$, and $\tilde{L}_{0}(C,Q)$, where $0<b<1$, $Q$ is a $d\times d$ matrix with eigenvalues having positive real parts, and $C$ is a closed multiplicative subsemigroup of $[0,1]$ containing 0 and 1. Further, each of these classes generates the Urbanik-Sato type decreasing sequence of its subclasses. Characterizations and relations of these classes and subclasses are established. They complement and generalize results of Bunge, Jurek, Maejima and Naito, and Sato and Yamazato.

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Makoto MAEJIMA. Ken-iti SATO. Toshiro WATANABE. "Operator Semi-Selfdecomposability, $(C,Q)$-Decomposability and Related Nested Classes." Tokyo J. Math. 22 (2) 473 - 509, December 1999. https://doi.org/10.3836/tjm/1270041450

Information

Published: December 1999
First available in Project Euclid: 31 March 2010

zbMATH: 0947.60010
MathSciNet: MR1727887
Digital Object Identifier: 10.3836/tjm/1270041450

Rights: Copyright © 1999 Publication Committee for the Tokyo Journal of Mathematics

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Vol.22 • No. 2 • December 1999
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