Discussiones Mathematicae Probability and Statistics 23(1) (2003) 77-81

ABOUT THE DENSITY OF SPECTRAL MEASURE OF THE TWO-DIMENSIONAL Sα S RANDOM  VECTOR

 Marta Borowiecka-Olszewska  and Jolanta K. Misiewicz

 Institute of Mathematics, University of Zielona Góra
Podgórna 50, 65-246 Zielona Góra, Poland
e-mail: m.borowiecka-olszewska@im.uz.zgora.pl
e-mail: j.misiewicz@im.uz.zgora.pl

Abstract

In this paper, we consider a symmetric α -stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function exp{ - (| a| p + | b| p)α /p } is a characteristic function of such a vector for some p and α . The solution of this problem we can find in [3], in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in [4].

Keywords: stable, sub-stable, maximal stable random vector, spectral measure.

References

[1]P. Billingsley, Probability and Measure, John Wiley & Sons, New York 1979.
[2]W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons, New York 1966.
[3]R. Grzaślewicz and J.K. Misiewicz, Isometric embeddings of subspaces of Lα -spaces and maximal representation for symmetric stable processes, Functional Analysis (1996), 179-182.
[4]J.K. Misiewicz and S. Takenaka, Some remarks on Sα S, β -sub-stable random vectors, preprint.
[5]J.K. Misiewicz, Sub-stable and pseudo-isotropic processes. Connections with the geometry of sub-spaces of Lα -spaces, Dissertationes Mathematicae CCCLVIII, 1996.
[6]J.K. Misiewicz and Cz. Ryll-Nardzewski, Norm dependent positive definite functions and measures on vector spaces, pp. 284-292 in: Probability Theory on Vector Spaces IV, a\'ncut 1987, Springer Verlag LNM 1391, 1989.
[7]G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, London 1993.

Received 7 April 2003