Probability and Mathematical Statistics: Vol. 25, Fasc. 2

THE FUNCTIONAL EQUATION AND STRICTLY SUBSTABLE RANDOM VECTORS

Marta Borowiecka-Olszewska

Abstract: A random vector $X$ is $b$ -substable, $b (- (0,2],$ if there exist a symmetric $b$ -stable random vector $Y$ and a random variable $Q > 0$ independent of $Y$ such that $d 1/b. X = YQ$ In this paper we investigate strictly $b$ -substable random vectors which are generated from a strictly $b$ -stable random vector $Y.$ We study some of their properties. We obtain the theorem that every strictly $b$ -stable random vector $X$ with $Q ~ Sa/b(s,1,0)$ is also strictly $a$ -stable, $a < b$ (for the case of random variable $X$ see, e.g., [1], [6]). The opposite theorem is also satisfied, but we obtain something more. We obtain some functional equation and we show that if a strictly $b$ -substable random vector $X$ is $a$ -stable, then it has to be strictly $a$ -stable and the mixing random variable $Q$ has to have a distribution $Sa/b(s,1,0).$ This is the main result of the paper.

2000 AMS Mathematics Subject Classification: 60A10, 60E07, 60E10.

Key words and phrases: Stable, strictly stable, substable random vectors, spectral measure, characteristic functions.