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The functional equation and strictly substable random vectors

Borowiecka-Olszewska M.

Probability and Mathematical Statistics

2005

Vol. 25, Fasc. 2

267--278

A random vector X is β-substable, β ∈ (0, 2), if there exist a symmetric β-stable random vector Y and a random variable Θ ≥ 0 independent of Y such that X [symbol] YΘ1/β. In this paper we investigate strictly β-substable random vectors which are generated from a strictly β-stable random vector Y. We study some of their properties. We obtain the theorem that every strictly β-stable random vector X with Θ ∼ Sα/β (σ, 1, 0) is also strictly α-stable, α < β (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 astrictly β-substable random vector X is α-stable, then it has to be strictly α-stable and the mixing random variable Θ has to have a distribution Sα/β (σ, 1, 0). This is the main result of the paper.