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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 22, Fasc. 2,
pages 253 - 258
 

SOME REMARKS ON SaS, b -SUBSTABLE RANDOM VECTORS

Jolanta K. Misiewicz
Shigeo Takenaka

Abstract: An SaS random vector X is b -substable, a < b < 2, if    d   1/b
X = Y Q  for some symmetric b -stable random vector Y,Q >  0 a random variable with the Laplace transform       a/b
exp(- t  ), Y and Q are independent. We say that an SaS random vector is maximal if it is not b -substable for any b > a.

In the paper we show that the canonical spectral measure for every SaS, b -substable random vector X, b > a, is equivalent to the Lebesgue measure on Sn-1. We show also that every such vector admits the representation X = Y + Z, where Y is an SaS sub-Gaussian random vector, Z is a maximal SaS random vector, Y and Z are independent. The last representation is not unique.

2000 AMS Mathematics Subject Classification: 60A99, 60E07, 60E10, 60E99.

Key words and phrases: Symmetric a -stable vector, substable distributions, spectral measure.

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