Cauchy transforms of measures viewed as some functionals of Fourier transforms

• Autorzy: Jurek Z. J.
• Języki publikacji: EN

Abstrakty

EN

The Cauchy transform of a positive measure plays an important role in complex analysis and more recently in so-called free probability. We show here that the Cauchy transform restricted to the imaginary axis can be viewed as the Fourier transform of some corresponding measures. Thus this allows the full use of that classical tool. Furthermore, we relate restricted Cauchy transforms to classical com- pound Poisson measures, exponential mixtures, geometric infinite divisibility and free-infinite divisibility. Finally, we illustrate our approach with some examples.

Słowa kluczowe

EN

Cauchy transform; Fourier transform; Laplace transform; compound Poisson distribution; Lévy process; random integral; infinitely divisible measure; geometric infinitely divisible measures; Voiculescu transform; free-infinitely divisible measures

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2006
• Tom: Vol. 26, Fasc. 1
• Strony: 187--200
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Jurek Z. J.

• Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

• [1] N. I. Akhiezer, The Classical Moment Problem, Oliver & Boyd, Edinburgh and London 1965.
• [2] A. Araujo and E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York 1980.
• [3] О. E. Barndorff-Nielsen and S. Thorbjornsen, Levy laws in free probability, Proc. Natl. Acad. Sci. USA 99 (26) (2002), pp. 16568-16575.
• [4] H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (3) (1993), pp. 733-773.
• [5] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statist. 76, Springer, New York 1992.
• [6] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th edition, Academic Press, San Diego 1994.
• [7] A. M. Iksanov, Z. J. Jurek and B. M. Schreiber, A new factorization property of the selfdecomposable probability measures, Ann. Probab. 32 (2) (2004), pp. 1356 -1369.
• [8] Z. J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures, Ann. Probab. 13 (2) (1985), pp. 592-608.
• [9] Z. J. Jurek, Random integral representation for classes of limit distributions similar to Levy class L0, Probab. Theory Related Fields 78 (1988), pp. 473-490.
• [10] Z. J. Jurek, On Levy (spectral) measures of integral form on Banach spaces, Probab. Math. Statist 11 (1) (1990), pp. 139-148.
• [11] Z. J. Jurek, Generalized Levy stochastic areas and selfdecomposability, Statist. Probab. Lett. 64 (2003), pp. 213-222.
• [12] Z. J. Jurek, Random integral representations for free-infinitely divisible and tempered stable distributions, preprint, 2004.
• [13] Z. J. Jurek and W. Vervaat, An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrscheinlichkeitstheorie verw. Gebiete 62 (1983), pp. 247-262.
• [14] L. B. Klebanov, G. M. Manija and I. A. Melamed, A problem of V M. Zolotarev and analogs of infinitely divisible and stable distributions in a scheme of random sums of random variables (in Russian), Teor. Veroyatnost. i Primenen. 29 (1984), pp. 757-760.
• [15] S. Lang, SL2(R), Addison-Wesley, Reading, Massachusetts, 1975.
• [16] B. M. Ramachandran, On geometric-stable laws, related property of stable processes, and stable densities of exponent one, Ann. Inst. Statist. Math. 49 (2) (1997), pp. 299-313.
• [17] R. Speicher and R. Woroudi, Boolean convolution, Fields Inst. Commun. 12 (1997), pp. 267-279.
• [18] D. W. Stroock, Probability Theory. An Analytic View, Cambridge University Press, 1994.
• [19] D. Voiculescu, Lectures on free probability theory, Lecture Notes in Math. No 1738 (1999), pp. 280-349. [Ecole d’Eté de Probab. de Saint-Flour, XXVIII, 1998.]