The functional equation and strictly substable random vectors

• Autorzy: Borowiecka-Olszewska M.
• Języki publikacji: EN

Abstrakty

EN

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] YΘ^1/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 ∼ S_α/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 S_α/β (σ, 1, 0). This is the main result of the paper.

Słowa kluczowe

EN

stable; strictly stable; substable random vectors; spectral measure; characteristic functions

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2005
• Tom: Vol. 25, Fasc. 2
• Strony: 267--278
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Borowiecka-Olszewska M.

• Department of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Podgórna 50, 65-246 Zielona Góra, Poland

Bibliografia

• [1] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York 1966.
• [2] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, Toronto 1953 (1st edition: 1937).
• [3] E. Lukács, Characteristic Functions, Griffin, London 1970.
• [4] J. K. Misiewicz, Sub-stable and pseudo-isotropic processes. Connections with the geometry of sub-spaces of Lα-spaces, Dissertationes Math. 358 (1996).
• [5] B. Ramachandran, Advanced Theory of Characteristic Functions, Statistical Publishing Society, Calcutta 1967.
• [6] G. Samorodnitsky and M. Taqqu, Stable Nan-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, London 1993.
• [7] K. Urbanik, Remarks on B-stable probability distributions, Bull. Acad. Polon. Sci. Math. 9 (1976), pp. 781-787.