Probability and Mathematical Statistics: Vol. 22, Fasc. 2

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.